Recent zbMATH articles in MSC 14https://zbmath.org/atom/cc/142021-11-25T18:46:10.358925ZWerkzeugA singular mathematical promenadehttps://zbmath.org/1472.000012021-11-25T18:46:10.358925Z"Ghys, Étienne"https://zbmath.org/authors/?q=ai:ghys.etienneAt the first look, one may feel that the book title is a little bit strange. The word singular in the title refers to the concept of singularity of a curve and does not mean a trip made by an individual person. It is a promenade into the mathematical world. The tour is interesting, entertaining and enjoyable, but it may be little bit difficult for those who have insufficient mathematical knowledge. So some mathematical maturity is required to fully appreciate the beauty presented by the author. When you go through the subjects of it you will find it a wonderfully crafted book. The book consists of 30 chapters. Each chapter provides a rich read. Several chapters are fairly independent from the rest of the book. It is a remarkable achievement in terms of its content, structure, and style. In almost all chapters the author shows excellent examples of mathematical exposition and utilize history to enrich a contemporary mathematical investigations. Actually he weaves historical stories in between the combinatorics, complex analysis, and algebraic geometry \dots etc. and does it all in a very readable and remarkable way. The design of the book is amazing: it contains many pictures and illustrations, scanned manuscripts, references, remarks, all written in the right margin of the pages (so one has the information immediately available). The text contains many historical quotations in different languages, with translations, and interesting analysis of the mathematics of our ``classics'' (Newton, Gauss, Hipparchus \dots etc). Hence the book will please any budding or professional mathematician. I can say that, principally, for professional readers, the book is an enjoyable reading due to the versatility of subjects using too many illustrations and remarks that enriched the concepts of the classical notions. In fact most of the material in the book can be regarded as an advanced undergraduate/early graduate level, even there are some material that is significantly more advanced. One very remarkable aspects of the book is the treat of historical matters. Some of the very classical notions such as the fundamental theorem of algebra, the theory of Puisseux series, the linking number of knots, discrete mathematics, operads, resolution of curve singularities, complex singularities, and more, have been discussed and explained in an enlightening way.
The author of the book, Professor Étienne Ghys, Director of Research at the École Normale Superiere de Lyon, is a skilled, gifted versatile expositor mathematician. He wrote his book in a relaxed, informal manner with lots of exclamation marks, figures, supporting computer graphics and illustrations that are mathematically helpful and visually engaging. It is interesting to know that most of illustrations have been produced by Ghys himself and who has waived all copyright and related or neighboring rights which is a good evidence of Ghys's service towards the dissemination of mathematical ideas. Ghys is a prominent researcher, broadly in geometry and dynamics. He was awarded the Clay Award for Dissemination of Mathematics in 2015.
As the author mentioned in his book, the motivation for writing such an interesting book came from a fact brought to his attention by his colleague, Maxim Kontsevich, in 2009 that relates the relative position of the graphs of four real polynomials under certain conditions imposed on the polynomials. So he begins the book with an attractive theorem of Maxim Kontsevich scribbled for him on a Paris metro ticket who stated it in a nice: Theorem. There do not exist four polynomials \(P_1, \dots , P_4 \in R[x] \) with \(P_1(x) < P_2(x) < P_3(x) < P_4(x)\) for all small negative \(x\), and \(P_2(x) < P_4(x) < P_1(x) < P_3(x)\) for small positive \(x\).
In fact Ghys begins his promenade with this attractive theorem. Amazingly, this result basically characterizes what can or cannot happen for crossings, not only for graphs of arbitrary collections of polynomials, but indeed for all real analytic planar curves. Actually the book explores very different questions related to this problem, and follows on different ramifications. Ghys discussed the more general singularities of algebraic curves in the plane, explaining how the concepts were developed historically. I recommend to assign parts of it as an independent studies for both undergraduate and graduate students.Book review of: G.-M. Greuel et al., Singular algebraic curveshttps://zbmath.org/1472.000122021-11-25T18:46:10.358925Z"Degtyarev, Alex"https://zbmath.org/authors/?q=ai:degtyarev.alexReview of [Zbl 1411.14001].Book review of: K. Fujiwara and F. Kato, Foundations of rigid geometry. Ihttps://zbmath.org/1472.000412021-11-25T18:46:10.358925Z"Wedhorn, Torsten"https://zbmath.org/authors/?q=ai:wedhorn.torstenReview of [Zbl 1400.14001].Some properties of the polynomially bounded o-minimal expansions of the real field and of some quasianalytic local ringshttps://zbmath.org/1472.030362021-11-25T18:46:10.358925Z"Berraho, M."https://zbmath.org/authors/?q=ai:berraho.mouradLet \(\mathcal E_n\) be the ring of germs of smooth functions in \(\mathbb R^n\) at the origin. The Borel map is a map from \(\mathcal E_n\) onto the ring of formal power series \(\mathbb R[[x_1,\ldots, x_n]]\). The image \(\widehat{f}\) of an \(f \in \mathcal E_n\) under the Borel map is defined as the infinite Taylor expansion of \(f\) at the origin. This paper discusses subrings \(\mathcal C\) of \(\mathcal E_n\) when the restrictions of the Borel map to \(\mathcal C\) are bijective. Its results are as follows:
Let \(\mathcal D_n\) be of the subring of \(\mathcal E_n\) consisting of the germs of real analytic functions definable in a polymonially bounded o-minimal expansion of the field of reals. The Weierstrass division and preparation theorems for \(\mathcal D_2\) hold true when the restriction of the Borel map to \(\mathcal D_1\) is bijective.
When a subring \(\mathcal C\) of \(\mathcal E_1\) satisfies the technical condition called the stability under monomial division and the restriction of the Borel map to it is bijective, the local ring \(\mathcal C\) with the \((x_1)\)-adic topology is complete.On framed simple purely real Hurwitz numbershttps://zbmath.org/1472.050072021-11-25T18:46:10.358925Z"Kazarian, M. E."https://zbmath.org/authors/?q=ai:kazaryan.maxim-e"Lando, S. K."https://zbmath.org/authors/?q=ai:lando.sergei-k"Natanzon, S. M."https://zbmath.org/authors/?q=ai:natanzon.sergei-mEnumerating coloured partitions in 2 and 3 dimensionshttps://zbmath.org/1472.050172021-11-25T18:46:10.358925Z"Davison, Ben"https://zbmath.org/authors/?q=ai:davison.ben"Ongaro, Jared"https://zbmath.org/authors/?q=ai:ongaro.jared"Szendrői, Balázs"https://zbmath.org/authors/?q=ai:szendroi.balazsSummary: We study generating functions of ordinary and plane partitions coloured by the action of a finite subgroup of the corresponding special linear group. After reviewing known results for the case of ordinary partitions, we formulate a conjecture concerning a basic factorisation property of the generating function of coloured plane partitions that can be thought of as an orbifold analogue of a conjecture of \textit{D. Maulik} et al. [Compos. Math. 142, No. 5, 1263--1285 (2006; Zbl 1108.14046)], now a theorem, in three-dimensional Donaldson-Thomas theory. We study natural quantisations of the generating functions arising from geometry, discuss a quantised version of our conjecture, and prove a positivity result for the quantised coloured plane partition function under a geometric assumption.The finite matroid-based valuation conjecture is falsehttps://zbmath.org/1472.050312021-11-25T18:46:10.358925Z"Tran, Ngoc Mai"https://zbmath.org/authors/?q=ai:tran.ngoc-maiGeneralisations of the Harer-Zagier recursion for 1-point functionshttps://zbmath.org/1472.051502021-11-25T18:46:10.358925Z"Chaudhuri, Anupam"https://zbmath.org/authors/?q=ai:chaudhuri.anupam"Do, Norman"https://zbmath.org/authors/?q=ai:do.normanSummary: \textit{J. Harer} and \textit{D. Zagier} [Invent. Math. 85, 457--485 (1986; Zbl 0616.14017)] proved a recursion to enumerate gluings of a \(2d\)-gon that result in an orientable genus \(g\) surface, in their work on Euler characteristics of moduli spaces of curves. Analogous results have been discovered for other enumerative problems, so it is natural to pose the following question: how large is the family of problems for which these so-called 1-point recursions exist? In this paper, we prove the existence of 1-point recursions for a class of enumerative problems that have Schur function expansions. In particular, we recover the Harer-Zagier recursion, but our methodology also applies to the enumeration of dessins d'enfant, to Bousquet-Mélou-Schaeffer numbers, to monotone Hurwitz numbers, and more. On the other hand, we prove that there is no 1-point recursion that governs single Hurwitz numbers. Our results are effective in the sense that one can explicitly compute particular instances of 1-point recursions, and we provide several examples. We conclude the paper with a brief discussion and a conjecture relating 1-point recursions to the theory of topological recursion.Crystal structures for symmetric Grothendieck polynomialshttps://zbmath.org/1472.051522021-11-25T18:46:10.358925Z"Monical, Cara"https://zbmath.org/authors/?q=ai:monical.cara"Pechenik, Oliver"https://zbmath.org/authors/?q=ai:pechenik.oliver"Scrimshaw, Travis"https://zbmath.org/authors/?q=ai:scrimshaw.travisSummary: The symmetric Grothendieck polynomials representing Schubert classes in the K theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type \(A_n\) crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate our crystal structure to combinatorial models using excited Young diagrams, Gelfand-Tsetlin patterns via the 5-vertex model, and biwords via Hecke insertion to compute symmetric Grothendieck polynomials.Crystal for stable Grothendieck polynomialshttps://zbmath.org/1472.051532021-11-25T18:46:10.358925Z"Morse, Jennifer"https://zbmath.org/authors/?q=ai:morse.jennifer"Pan, Jianping"https://zbmath.org/authors/?q=ai:pan.jianping"Poh, Wencin"https://zbmath.org/authors/?q=ai:poh.wencin"Schilling, Anne"https://zbmath.org/authors/?q=ai:schilling.anneA crystal is a directed graph whose encoded information mirror that of the highest weight theory of a root system. Their importance relies on that it reduces problems about representations of Kac-Moody Lie algebras to analogous problems but in a purely combinatorial context; and conversely. References introducing crystals are, from a combinatorial point of view, [\textit{D. Bump} and \textit{A. Schilling}, Crystal bases. Representations and combinatorics. Hackensack, NJ: World Scientific (2017; Zbl 1440.17001); \textit{P. Hersh} and \textit{C. Lenart}, Math. Z. 286, No. 3--4, 1435--1464 (2017; Zbl 1371.05315)]; from the algebraic side, see [\textit{J. Hong} and \textit{S.-J. Kang}, Introduction to quantum groups and crystal bases. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1134.17007)].
Here, the authors associate a type A crystal on the set of \(321\)-avoiding Hecke factorizations; for an expanded version of the content of this paper see [\textit{J. Morse} et al., Electron. J. Comb. 27, No. 2, Research Paper P2.29, 48 p. (2020; Zbl 1441.05237)]. More references: [\textit{M. Albert} et al., Eur. J. Comb. 78, 44--72 (2019; Zbl 1414.05004); \textit{M. Bóna}, Combinatorics of permutations. Boca Raton, FL: CRC Press (2012; Zbl 1255.05001)]. The authors also define a new insertion from decreasing factorizations to pairs of semistandard Yount tableaux, and prove several properties; in particular, this new insertion intertwines with the crystal operators. Everything is related with the combinatorics of Young tableaux.
Additional references: [\textit{M. Gillespie} et al., Algebr. Comb. 3, No. 3, 693--725 (2020; Zbl 1443.05183); \textit{Y.-T. Oh} and \textit{E. Park}, Electron. J. Comb. 26, No. 4, Research Paper P4.39, 19 p. (2019; Zbl 1428.05329); \textit{S. Assaf} and \textit{E. K. Oğuz}, Sémin. Lothar. Comb. 80B, 80B.26, 12 p. (2018; Zbl 1411.05272); \textit{J.-H. Kwon}, Handb. Algebra 6, 473--504 (2009; Zbl 1221.17017); \textit{G. Benkart} and \textit{S.-J. Kang}, Adv. Stud. Pure Math. 28, 21--54 (2000; Zbl 1027.17009); \textit{T. H. Baker}, Prog. Math. 191, 1--48 (2000; Zbl 0974.05080); \textit{G. Cliff}, J. Algebra 202, No. 1, 10--35 (1998; Zbl 0969.17010); \textit{P. Littelmann}, J. Algebra 175, No. 1, 65--87 (1995; Zbl 0831.17004); \textit{A. Puskás}, Assoc. Women Math. Ser. 16, 333--362 (2019; Zbl 1416.05300); \textit{V. I. Danilov} et al., Algebra 2013, Article ID 483949, 14 p. (2013; Zbl 1326.05045); \textit{T. Lam} and \textit{P. Pylyavskyy}, Sel. Math., New Ser. 19, No. 1, 173--235 (2013; Zbl 1260.05043); \textit{V. Genz} et al., Sel. Math., New Ser. 27, No. 4, Paper No. 67, 45 p. (2021; Zbl 07383344); \textit{N. Jacon}, Electron. J. Comb. 28, No. 2, Research Paper P2.21, 16 p. (2021; Zbl 07356164); \textit{T. Shoji} and \textit{Z. Zhou}, J. Algebra 569, 67--110 (2021; Zbl 07286477)].Alcove paths and Gelfand-Tsetlin patternshttps://zbmath.org/1472.051542021-11-25T18:46:10.358925Z"Watanabe, Hideya"https://zbmath.org/authors/?q=ai:watanabe.hideya"Yamamura, Keita"https://zbmath.org/authors/?q=ai:yamamura.keitaSummary: In their study of the equivariant K-theory of the generalized flag varieties \(G/P\), where \(G\) is a complex semisimple Lie group, and \(P\) is a parabolic subgroup of \(G\), \textit{C. Lenart} and \textit{A. Postnikov} [Trans. Am. Math. Soc. 360, No. 8, 4349--4381 (2008; Zbl 1211.17021)] introduced a combinatorial tool, called the alcove path model. It provides a model for the highest weight crystals with dominant integral highest weights, generalizing the model by semistandard Young tableaux. In this paper, we prove a simple and explicit formula describing the crystal isomorphism between the alcove path model and the Gelfand-Tsetlin pattern model for type \(A\).The Bruhat order, the lookup conjecture and spiral Schubert varieties of type \(\tilde{A}_2\)https://zbmath.org/1472.051552021-11-25T18:46:10.358925Z"Graham, William"https://zbmath.org/authors/?q=ai:graham.william-a"Li, Wenjing"https://zbmath.org/authors/?q=ai:li.wenjingSummary: Although the Bruhat order on a Weyl group is closely related to the singularities of the Schubert varieties for the corresponding Kac-Moody group, it can be difficult to use this information to prove general theorems. This paper uses the action of the affine Weyl group of type \(\tilde{A}_2\) on a Euclidean space \(V \cong \mathbb{R}^2\) to study the Bruhat order on \(W\). We believe that these methods can be used to study the Bruhat order on arbitrary affine Weyl groups. Our motivation for this study was to extend the lookup conjecture of \textit{B. D. Boe} and \textit{W. Graham} [Am. J. Math. 125, No. 2, 317--356 (2003; Zbl 1074.14045)] (which is a conjectural simplification of the Carrell-Peterson criterion (see [\textit{J. B. Carrell}, Proc. Symp. Pure Math. 56, 53--61 (1994; Zbl 0818.14020)]) for rational smoothness) to type \(\tilde{A}_2\). Computational evidence suggests that the only Schubert varieties in type \(\tilde{A}_2\) where the ``nontrivial'' case of the lookup conjecture occurs are the spiral Schubert varieties, and as a step towards the lookup conjecture, we prove it for a spiral Schubert variety \(X ( w )\) of type \(\tilde{A}_2\). The proof uses descriptions we obtain of the elements \(x \leq w\) and of the rationally smooth locus of \(X ( w )\) in terms of the \(W\)-action on \(V\). As a consequence we describe the maximal nonrationally smooth points of \(X ( w )\). The results of this paper are used in a sequel to describe the smooth locus of \(X ( w )\), which is different from the rationally smooth locus.Algebraic geometry for \(\ell \)-groupshttps://zbmath.org/1472.060232021-11-25T18:46:10.358925Z"Di Nola, Antonio"https://zbmath.org/authors/?q=ai:di-nola.antonio"Lenzi, Giacomo"https://zbmath.org/authors/?q=ai:lenzi.giacomo"Vitale, Gaetano"https://zbmath.org/authors/?q=ai:vitale.gaetanoSummary: In this paper we focus on the algebraic geometry of the variety of \(\ell \)-groups (i.e. lattice ordered abelian groups). In particular we study the role of the introduction of constants in functional spaces and \(\ell \)-polynomial spaces, which are themselves \(\ell \)-groups, evaluated over other \(\ell \)-groups. We use different tools and techniques, with an increasing level of abstraction, to describe properties of \(\ell \)-groups, topological spaces (with the Zariski topology) and a formal logic, all linked by the underlying theme of solutions of \(\ell \)-equations.Erratum to: ``Geometric aspects of Lucas sequences. I''https://zbmath.org/1472.110712021-11-25T18:46:10.358925Z"Suwa, Noriyuki"https://zbmath.org/authors/?q=ai:suwa.noriyukiCorrects mistakes in the statements of Corollary 3.13 and Corollary 3.14 in [the author, ibid. 43, No. 1, 75--136 (2020; Zbl 1469.11026)].Retract rationality and algebraic torihttps://zbmath.org/1472.111122021-11-25T18:46:10.358925Z"Scavia, Federico"https://zbmath.org/authors/?q=ai:scavia.federicoSummary: For any prime number \(p\) and field \(k\), we characterize the \(p\)-retract rationality of an algebraic \(k\)-torus in terms of its character lattice. We show that a \(k\)-torus is retract rational if and only if it is \(p\)-retract rational for every prime \(p\), and that the Noether problem for retract rationality for a group of multiplicative type \(G\) has an affirmative answer for \(G\) if and only if the Noether problem for \(p\)-retract rationality for \(G\) has a positive answer for all \(p\). For every finite set of primes \(S\) we give examples of tori that are \(p\)-retract rational if and only if \(p\notin S\).Adic shtukas, modifications and applicationshttps://zbmath.org/1472.111602021-11-25T18:46:10.358925Z"Hieu, Nguyen Kieu"https://zbmath.org/authors/?q=ai:hieu.nguyen-kieuSummary: In this paper, via the study of the modifications of vector bundles on the Fargues-Fontaine curve, we prove a geometric formula relating the Lubin-Tate towers with the simple basic unramified Rapoport-Zink spaces of EL type of signature \((1, n-1), (p_1, q_1), \dots, (p_k, q_k)\) where \(p_iq_i = 0 \). In particular, we deduce the computation of the cohomology groups of the latter.The abelian part of a compatible system and \(\ell \)-independence of the Tate conjecturehttps://zbmath.org/1472.111682021-11-25T18:46:10.358925Z"Hui, Chun Yin"https://zbmath.org/authors/?q=ai:hui.chun-yinSummary: Let \(K\) be a number field and \(\{V_\ell \}_\ell\) a rational strictly compatible system of semisimple Galois representations of \(K\) arising from geometry. Let \(\mathbf{G}_\ell\) and \(V_\ell^{{\text{ab}}}\) be respectively the algebraic monodromy group and the maximal abelian subrepresentation of \(V_\ell\) for all \(\ell \). We prove that the system \(\{V_\ell^{{\text{ab}}}\}_\ell\) is also a rational strictly compatible system under some group theoretic conditions, e.g., when \(\mathbf{G}_{\ell '}\) is connected and satisfies \textit{Hypothesis A} for some prime \(\ell '\). As an application, we prove that the Tate conjecture for abelian variety \(X/K\) is independent of \(\ell\) if the algebraic monodromy groups of the Galois representations of \(X\) satisfy the required conditions.Versal families of elliptic curves with rational 3-torsionhttps://zbmath.org/1472.111742021-11-25T18:46:10.358925Z"Bekker, Boris M."https://zbmath.org/authors/?q=ai:bekker.boris-m"Zarhin, Yuri G."https://zbmath.org/authors/?q=ai:zarhin.yuri-gTorsion of rational elliptic curves over the maximal abelian extension of \(\mathbb{Q} \)https://zbmath.org/1472.111762021-11-25T18:46:10.358925Z"Chou, Michael"https://zbmath.org/authors/?q=ai:chou.michaelSummary: Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\), and let \(\mathbb{Q}^{\mathrm{ab}}\) be the maximal abelian extension of \(\mathbb {Q}\). In this article we classify the groups that can arise as \(E(\mathbb{Q}^{\mathrm{ab}})_{\operatorname{tors}}\) up to isomorphism. The method illustrates techniques for finding explicit models of modular curves of mixed level structure. Moreover, we provide an explicit algorithm to compute \(E(\mathbb{Q}^{\mathrm{ab}})_{\operatorname{tors}}\) for any elliptic curve \(E/\mathbb{Q}\).Average values on the Jacobian variety of a hyperelliptic curvehttps://zbmath.org/1472.111772021-11-25T18:46:10.358925Z"Chung, Jiman"https://zbmath.org/authors/?q=ai:chung.jiman"Im, Bo-Hae"https://zbmath.org/authors/?q=ai:im.bo-haeSummary: We give explicitly an average value formula under the multiplication-by-2 map for the \(x\)-coordinates of the 2-division points \(D\) on the Jacobian variety \(J(C)\) of a hyperelliptic curve \(C\) with genus \(g\) if \(2D \equiv 2P-2\infty \pmod{\text{Pic}(C)}\) for \(P=(x_P, y_P) \in C\) with \(y_P \ne 0\). Moreover, if \(g=2\), we give a more explicit formula for \(D\) such that \(2D \equiv P-\infty \pmod{\mathrm{Pic}(C)}\).On elliptic curves of prime power conductor over imaginary quadratic fields with class number 1https://zbmath.org/1472.111782021-11-25T18:46:10.358925Z"Cremona, John"https://zbmath.org/authors/?q=ai:cremona.john-e"Pacetti, Ariel"https://zbmath.org/authors/?q=ai:pacetti.arielSummary: The main result of this paper is to extend from \(\mathbb Q\) to each of the nine imaginary quadratic fields of class number 1 a result of [\textit{J.-P. Serre}, Duke Math. J. 54, 179--230 (1987; Zbl 0641.10026)] and [\textit{J. F. Mestre} and \textit{J. Oesterlé}, J. Reine Angew. Math. 400, 173--184 (1989; Zbl 0693.14004)], namely that if \(E\) is an elliptic curve of prime conductor, then either \(E\) or a 2-, 3- or 5-isogenous curve has prime discriminant. For four of the nine fields, the theorem holds with no change, while for the remaining five fields the discriminant of a curve with prime conductor is (up to isogeny) either prime or the square of a prime. The proof is conditional in two ways: first that the curves are modular, so are associated to suitable Bianchi newforms; and second that a certain level-lowering conjecture holds for Bianchi newforms. We also classify all elliptic curves of prime power conductor and non-trivial torsion over each of the nine fields: in the case of 2-torsion, we find that such curves either have CM or with a small finite number of exceptions arise from a family analogous to the Setzer-Neumann family over \(\mathbb Q\).Explicit moduli spaces for congruences of elliptic curveshttps://zbmath.org/1472.111792021-11-25T18:46:10.358925Z"Fisher, Tom"https://zbmath.org/authors/?q=ai:fisher.tom-aIn the paper under review, the author determines explicit birational models over \({\mathbb Q}\) for the modular surfaces parametrizing pairs of \(N\)-congruent elliptic curves in all cases when this surface is an elliptic surface. In each case, the author also determines the rank of the Mordell-Weil lattice and the geometric Picard number.
More precisely, for an integer \(N\geq 2\), two elliptic curves are said to be \(N\)-congruent if their \(N\)-torsion subgroups are isomorphic as Galois modules. Such an isomorphism raises the Weil pairing to the power \(\varepsilon\) for some \(\varepsilon\in ({\mathbb Z}/N{\mathbb Z})^\times\). In this case, one says that the \(N\)-congruence has power \(\varepsilon.\) Note that the author considers \(\varepsilon\) up to a square, since multiplication by \(m\) (with \(\gcd(m,N)=1\)) on one of the elliptic curves changes \(\varepsilon\) to \(m^2\varepsilon.\) Let \(Z(N,\varepsilon)\) be the surface parametrizing pairs of elliptic curves with power \(\varepsilon\), up to simultaneous quadratic twist. This surface is defined over \({\mathbb Q}\). Refining the previous classification of \textit{E. Kani} and \textit{W. Schanz} [Math. Z. 227, No. 2, 337--366 (1998; Zbl 0996.14012)] who explicitly determined the pairs \((N,\varepsilon)\) for which \(Z(N,\varepsilon)\) is birational over \({\mathbb C}\), the author shows that both the cases of an elliptic \(K3\)-surface and of an elliptic surface with Kodaira dimension one (a.k.a. a properly elliptic surface) are birational over \({\mathbb Q}\) to an elliptic surface. Furthermore, the author determines in each case a Weierstrass equation for the generic fibre as an elliptic curve over \({\mathbb Q}(T).\) Note that the author considers an elliptic surface to have a section. The explicit models are given as follows. The author noted that some of the cases were already treated in [\textit{Z. Chen}, Math. Proc. Camb. Philos. Soc. 165, No. 1, 137--162 (2018; Zbl 1451.11049); \textit{T. Fisher}, Acta Arith. 171, No. 4, 371--387 (2015; Zbl 1341.11028); \textit{A. Kumar}, Res. Math. Sci. 2, Paper No. 24, 46 p. (2015; Zbl 1380.11049)].
Theorem 1.1. The surfaces \(Z(N,\varepsilon)\) that are birational over \({\mathbb C}\) to an elliptic \(K3\)-surface, are in fact birational over \({\mathbb Q}\) to an elliptic surface. The generic fibres are the elliptic curves over \({\mathbb Q}(T)\) with the following Weierstrass equations.
\[
\begin{split}
Z(6,5)&:\quad y^2+3T(T-2)xy+2(T-1)(T+2)^2(T^3-2)y=x^3-6(T-1)(T^3-2)x^2,\\
Z(7,3)&:\quad y^2=x^3+(4T^4+4T^3-51T^2-2T-50)x^2+(6T+25)(52T^2-4T+25)x,\\
Z(8,3)&:\quad y^2=x^3-(3T^2-7)x^2-4T^2(4T^4-15)x+4T^2(53T^4+81T^2+162),\\
Z(8,5)&:\quad y^2=x^3-2(T^2+19)x^2-(4T^2-49)(T^4-6T^2+25)x,\\
Z(9,1)&:\quad y^2+(6T^2+3T+2)xy+T^2(T+1)(4T^3+9T+9)y+ x^3-(16T^4+12T^3+9T^2+6T+1)x^2,\\
Z(12,1)&:\quad y^2+2(5T^2+9)xy+96(T^2+3)(T^2+1)^2y+x^3+(T^2+3)(11T^2+1)x^2.
\end{split}
\]
Theorem 1.2. The surfaces \(Z(N,\varepsilon)\) that are birational over \({\mathbb C}\) to a properly elliptic surface, are in fact birational over \({\mathbb Q}\) to an elliptic surface. The generic fibres are the elliptic curves over \({\mathbb Q}(T)\) with the following Weierstrass equations.
\[
\begin{split}
Z(8,7)&:\quad y^2=x^3+2(4T^6-15T^4+14T^2-1)x^2+(T^2-1)^4(16T^4-24T^2+1)x,\\
Z(9,2)&:\quad y^2+3(4T^3+T^2-2)xy+(T-1)^3(T^3-1)(4T^3-3T-7)y+ x^3\\ & \quad -3(T+1)(T^3-1)(9T^2+2T+1)x^2,\\
Z(10,1)&:\quad y^2-(3T-2)(6T^2-5T-2)xy -4T^2(T-1)^2(4T^2-2T-1)(27T^3-54T^2+16T+12)y\\ & \quad +x^3+T^2(T-1)(27T^3-54T^2+16T+12)x^2,\\
Z(10,3)&:\quad y^2+(T^3-8T^2-9T-8)xy+2T^2(T^3-T^2-3T-3)(7T^2+2T+3)y\\ & \quad +x^3+2(3T+2)(T^3-T^2-3T-3)x^2,\\
Z(11,1)&:\quad y^2+(T^3+T)xy=x^3-(4T^5-17T^4+30T^3-18T^2+4)x^2 +T^2(2T-1)(3T^2-7T+5)^2x
\end{split}
\]
Using these explicit equations, the author applies the methods of van Luijk and Kloosterman to compute the geometric Picard number of each surface [\textit{R. Van Luijk}, J. Number Theory 123, No. 1, 92--119 (2007; Zbl 1160.14029); \textit{R. van Luijk}, Algebra Number Theory 1, No. 1, 1--17 (2007; Zbl 1123.14022); \textit{R. Kloosterman}, Can. Math. Bull. 50, No. 2, 215--226 (2007; Zbl 1162.14024)].Bounding cubic-triple product Selmer groups of elliptic curveshttps://zbmath.org/1472.111822021-11-25T18:46:10.358925Z"Liu, Yifeng"https://zbmath.org/authors/?q=ai:liu.yifengLet \(F\) be a totally real cubic number field, let \(E\) be a modular elliptic curve defined over \(F\) and let \(h^1(E)\) be its \(F\)-motive. Via multiplicative induction to \(\mathbb{Q}\) one gets a \textit{cubic-triple product motive} \(\mathrm{M}(E):=(\otimes\mathrm{Ind}^F_{\mathbb{Q}} h^1(E))(2)\) of dimension 8, whose \(p\)-adic realization is basically (a twist of) the multiplicative induction from \(F\) to \(\mathbb{Q}\) of the \(p\)-adic Tate module of \(E\). To such an object one can attach a triple product \(L\)-function \(L(s,\mathrm{M}(E))\) with good meromorphic properties and a functional equation with central critical value in \(s=0\).
The paper deals with an instance of the Bloch-Kato conjecture for this setting: in particular, under some additional hypotheses on \(E\), it proves that the nonvanishing of \(L(0,\mathrm{M}(E))\) yields the 0-dimensionality (over \(\mathbb{Q}_p\)) of the \(p\)-part of the appropriate Selmer group for infinitely many primes \(p\). The author already presented a similar result, where \(F\) was replaced by the product of \(\mathbb{Q}\) and a real quadratic field, in [\textit{Y. Lin}, Invent. Math. 205, No. 3, 693--780 (2016; Zbl 1395.11091)]: the strategy is based on a reciprocity law for cycles on a triple product of modular curves, which is obtained via congruence formulas arising from computations of étale local cohomology groups of varieties. Though the strategy is similar to the Lin paper mentioned above, the different setting requires new techniques on weight spectral sequences to handle higher dimensional cycles, which then provide the cohomological computations (hence the reciprocity law) needed here.
When \(L(0,\mathrm{M}(E))\neq 0\), the reciprocity law on some special (Hirzebruch-Zagier) cycles produces enough annihilators for the Selmer group to prove its finiteness.The elliptic KZB connection and algebraic de Rham theory for unipotent fundamental groups of elliptic curveshttps://zbmath.org/1472.111832021-11-25T18:46:10.358925Z"Luo, Ma"https://zbmath.org/authors/?q=ai:luo.ma.1Summary: We develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of \textit{D. Calaque} et al. [Prog. Math. 269, 165--266 (2009; Zbl 1241.32011)] and \textit{A. Levin} and \textit{G. Racinet} [``Towards multiple elliptic polylogarithms'', Preprint, \url{arXiv:math/0703237}]. We use it to give an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity.Automorphism groups of simple polarized abelian varieties of odd prime dimension over finite fieldshttps://zbmath.org/1472.111852021-11-25T18:46:10.358925Z"Hwang, WonTae"https://zbmath.org/authors/?q=ai:hwang.wontae.1|hwang.wontae.2Summary: We prove that the automorphism groups of simple polarized abelian varieties of odd prime dimension over finite fields are cyclic, and give a complete list of finite groups that can be realized as such automorphism groups.Arithmetical power series expansion of the sigma function for a plane curvehttps://zbmath.org/1472.111862021-11-25T18:46:10.358925Z"Ônishi, Yoshihiro"https://zbmath.org/authors/?q=ai:onishi.yoshihiroSummary: The Weierstrass function \(\sigma(u)\) associated with an elliptic curve can be generalized in a natural way to an entire function associated with a higher genus algebraic curve. This generalized multivariate sigma function has been investigated since the pioneering work of Felix Klein. The present paper shows Hurwitz integrality of the coefficients of the power series expansion around the origin of the higher genus sigma function associated with a certain plane curve, which is called an \((n, s)\)-curve or a plane telescopic curve. For the prime (2), the expansion of the sigma function is not Hurwitz integral, but its square is. This paper clarifies the precise structure of this phenomenon. In Appendix A, computational examples for the trigonal genus 3 curve ((3, 4)-curve) \(y^3+(\mu_1 x+\mu_4)y^2 + (\mu_2 x^2 + \mu_5 x + \mu_8)y = x^4 + \mu_3 x^3 + \mu_6 x^2 + \mu_9 x + \mu_{12}\) (where \(\mu_j\) are constants) are given.Twisted arithmetic Siegel Weil formula on \(X_{0}(N)\)https://zbmath.org/1472.111882021-11-25T18:46:10.358925Z"Du, Tuoping"https://zbmath.org/authors/?q=ai:du.tuoping"Yang, Tonghai"https://zbmath.org/authors/?q=ai:yang.tonghaiSummary: In this paper, we study twisted arithmetic divisors on the modular curve \(\mathcal{X}_0(N)\)with \(N\) square-free. For each pair \(({\Delta}, r)\), where \({\Delta} \equiv r^2 \operatorname{mod} 4 N\)and \({\Delta}\)is a fundamental discriminant, we construct a twisted arithmetic theta function \(\hat{\phi}_{{\Delta}, r}(\tau)\)which is a generating function of arithmetic twisted Heegner divisors. We prove that the arithmetic pairing \(\langle \hat{\phi}_{{\Delta}, r}(\tau), \hat{\omega}_N \rangle\)is equal to the special value, rather than the derivative, of some Eisenstein series, thanks to some cancellation, where \(\hat{\omega}_N\)is a normalized metric Hodge line bundle. We also prove the modularity of \(\hat{\phi}_{{\Delta}, r}(\tau)\).On Faltings heights of abelian varieties with complex multiplicationhttps://zbmath.org/1472.111892021-11-25T18:46:10.358925Z"Yuan, Xinyi"https://zbmath.org/authors/?q=ai:yuan.xinyiSummary: This expository article introduces some conjectures and theorems related to the Faltings heights of abelian varieties with complex multiplication. The topics include the Colmez conjecture, the averaged Colmez conjecture, and the André-Oort conjecture.
For the entire collection see [Zbl 1423.00001].On the level of modular curves that give rise to isolated \(j\)-invariantshttps://zbmath.org/1472.111902021-11-25T18:46:10.358925Z"Bourdon, Abbey"https://zbmath.org/authors/?q=ai:bourdon.abbey"Ejder, Özlem"https://zbmath.org/authors/?q=ai:ejder.ozlem"Liu, Yuan"https://zbmath.org/authors/?q=ai:liu.yuan"Odumodu, Frances"https://zbmath.org/authors/?q=ai:odumodu.frances"Viray, Bianca"https://zbmath.org/authors/?q=ai:viray.biancaSummary: We say a closed point \(x\) on a curve \(C\) is sporadic if \(C\) has only finitely many closed points of degree at most \(\deg(x)\) and that \(x\) is isolated if it is not in a family of effective degree \(d\) divisors parametrized by \(\mathbb{P}^1\) or a positive rank abelian variety (see Section 4 for more precise definitions and a proof that sporadic points are isolated). Motivated by well-known classification problems concerning rational torsion of elliptic curves, we study sporadic and isolated points on the modular curves \(X_1(N)\). In particular, we show that any non-cuspidal non-CM sporadic, respectively isolated, point \(x \in X_1(N)\) maps down to a sporadic, respectively isolated, point on a modular curve \(X_1(d)\), where \(d\) is bounded by a constant depending only on \(j(x)\). Conditionally, we show that \(d\) is bounded by a constant depending only on the degree of \(\mathbb{Q}(j(x))\), so in particular there are only finitely many \(j\)-invariants of bounded degree that give rise to sporadic or isolated points.Quadratic Chabauty for modular curves and modular forms of rank onehttps://zbmath.org/1472.111912021-11-25T18:46:10.358925Z"Dogra, Netan"https://zbmath.org/authors/?q=ai:dogra.netan"Le Fourn, Samuel"https://zbmath.org/authors/?q=ai:le-fourn.samuelThe Chabauty-Kim method is a method for determining the set \(X(\mathbb{Q})\) of rational points of a curve \(X\) over \(\mathbb{Q}\) of genus bigger than one. The idea is to locate \(X(\mathbb{Q})\) inside \(X(\mathbb{Q}_p)\) by finding an obstruction to a \(p\)-adic point being global. This leads to a tower of obstructions \[X(\mathbb{Q}_p) \supset X(\mathbb{Q}_p)_1 \supset X(\mathbb{Q}_p)_2 \supset \ldots \supset X(\mathbb{Q}).\] The first obstruction set \(X(\mathbb{Q}_p)_1\) is the one produced by Chabauty's method. In situations when \(X(\mathbb{Q}_p)_1\) is finite, it can often be used to determine \(X(\mathbb{Q})\).
In the present paper, the authors study the finiteness of the Chabauty-Kim set \(X(\mathbb{Q}_p)_2\) when \(X\) is one of the modular curves \(X_{\text{ns}}^{+}(N)\) or \(X_0^{+}(N)\) with \(N\) a prime different from \(p\), where \(X_0^{+}(N)\) is the quotient of \(X_0(N)\) by the Atkin-Lehner involution \(w_N\), and \(X_{\text{ns}}^{+}(N)\) is the quotient of \(X(N)\) by the normalizer of a non-split Cartan subgroup. They show that for all prime \(N\) such that \(g(X_0^{+}(N)) \geq 2\), \(X_0^{+}(N)(\mathbb{Q}_p)_2\) is finite for any \(p \neq N\), and for all prime \(N\) such that \(g(X_{\text{ns}}^{+}(N)) \geq 2\) and \(X_{\text{ns}}^{+}(N)(\mathbb{Q}) \neq \emptyset\), \(X_{\text{ns}}^{+}(N)(\mathbb{Q}_p)_2\) is finite for any \(p \neq N\). Their proof proceeds along the lines of the quadratic Chabauty method.Ordinary points \(\bmod\, p\) of \(\operatorname{GL}_n(\mathbb{R})\)-locally symmetric spaceshttps://zbmath.org/1472.111922021-11-25T18:46:10.358925Z"Goresky, Mark"https://zbmath.org/authors/?q=ai:goresky.mark"Tai, Yung sheng"https://zbmath.org/authors/?q=ai:tai.yungshengSummary: Locally symmetric spaces for \(\operatorname{GL}_n(\mathbb{R})\) parametrize polarized complex abelian varieties with real structure (antiholomorphic involution). We introduce a mod \(p\) analog. We define an ``antiholomorphic'' involution (or ``real structure'') on an ordinary abelian variety (defined over a finite field \(k)\) to be an involution of the associated Deligne module \((T,F,V)\) that exchanges \(F\) (the Frobenius) with \(V\) (the Verschiebung). The definition extends to include principal polarizations and level structures. We show there are finitely many isomorphism classes of such objects in each dimension, and give a formula for this number that resembles the Kottwitz ``counting formula'' (for the number of principally polarized abelian varieties over \(k)\), but the symplectic group in the Kottwitz formula has been replaced by the general linear group.Cohomology of automorphic bundleshttps://zbmath.org/1472.111932021-11-25T18:46:10.358925Z"Lan, Kai-Wen"https://zbmath.org/authors/?q=ai:lan.kai-wenSummary: In this survey article, we review some recent works (by the author and his collaborators Junecue Suh, Michael Harris, Richard Taylor, Jack Thorne, and Benoît Stroh) on the cohomology of automorphic bundles overlocally symmetric varieties and some related geometric objects.
For the entire collection see [Zbl 1423.00001].Fourier-Jacobi cycles and arithmetic relative trace formulahttps://zbmath.org/1472.111942021-11-25T18:46:10.358925Z"Liu, Yifeng"https://zbmath.org/authors/?q=ai:liu.yifengSummary: In this article, we develop an arithmetic analogue of Fourier-Jacobi period integrals for a pair of unitary groups of equal rank. We construct the so-called Fourier-Jacobi cycles, which are algebraic cycles on the product of unitary Shimura varieties and abelian varieties. We propose the arithmetic Gan-Gross-Prasad conjecture for these cycles, which is related to the central derivatives of certain Rankin-Selberg \(L\)-functions, and develop a relative trace formula approach toward this conjecture.On Shimura varieties for unitary groupshttps://zbmath.org/1472.111952021-11-25T18:46:10.358925Z"Rapoport, M."https://zbmath.org/authors/?q=ai:rapoport.michael"Smithling, B."https://zbmath.org/authors/?q=ai:smithling.brian-d"Zhang, W."https://zbmath.org/authors/?q=ai:zhang.wei.1In the present paper, the authors define a class of Shimura varieties closely related to unitary groups which represent a moduli problem of abelian varieties with additional structure. This class of Shimura varieties is a variant of the Deligne-Kottwitz Shimura varieties. They compare their Shimura varieties with other unitary Shimura varieties.Uniform bounds for periods of endomorphisms of varietieshttps://zbmath.org/1472.111972021-11-25T18:46:10.358925Z"Huang, Keping"https://zbmath.org/authors/?q=ai:huang.kepingLet \(K\) be a finite extension of \({\mathbb Q}_{p}\) and \(R\) be its ring of integers. Assume that \(X\) is an algebraic variety, defined as the common zeroes of polynomials with coefficients in \(R\), and that \(f:X \to X\) is an endomorphim, also defined over \(R\).
A point \(P \in X\) is a periodic point of \(f\) if there is a positive integer \(n\) with \(f^{n}(P)=P\). The minimal of these integers \(n\) is called the primitive period of \(P\).
Let \(P \in X(R)\) (i.e, a \(R\)-rational point of \(X\)) be a periodic point with primitive period \(n\). The main result of this paper is an explicit upper bound for \(n\) in terms of the residue field of \(K\) and the valuation of \(K\). The proof is short (just two pages).
This result is related to Morton-Silverman's Conjecture on the existence of an upper bound \(C(D,N,d)\) for the cardinality of preperiodic \(K\)-rational points of \(f\) in the case that \(K\) is a finite extesion of degree \(D\) of \({\mathbb Q}\), \(X={\mathbb P}^{N}\) and \(f\) has degree \(d\).An addendum to the elliptic torsion anomalous conjecture in codimension 2https://zbmath.org/1472.111992021-11-25T18:46:10.358925Z"Hubschmid, Patrik"https://zbmath.org/authors/?q=ai:hubschmid.patrik"Viada, Evelina"https://zbmath.org/authors/?q=ai:viada.evelinaSummary: The torsion anomalous conjecture states that for any variety \(V\) in an abelian variety there are only finitely many maximal \(V\)-torsion anomalous varieties. We prove this conjecture for \(V\) of codimension 2 in a product \(E^N\) of an elliptic curve \(E\) without CM, complementing previous results for \(E\) with CM. We also give an effective upper bound for the normalized height of these maximal \(V\)-torsion anomalous varieties.The distribution relation and inverse function theorem in arithmetic geometryhttps://zbmath.org/1472.112002021-11-25T18:46:10.358925Z"Matsuzawa, Yohsuke"https://zbmath.org/authors/?q=ai:matsuzawa.yohsuke"Silverman, Joseph H."https://zbmath.org/authors/?q=ai:silverman.joseph-hillelThe paper presents an arithmetic distribution relation as well as two versions of the inverse function theorem in terms of an arithmetic distance function. In sections 3-5, the field \(K\) denotes a field with a complete set of absolute values \(\mathcal{M}_K\) satisfying a product formula. If we fix an algebraic closure \(\bar{K}\) of \(K\), an \(\mathcal{M}_K\)-constant will be a function \(\gamma \colon \mathcal{M}_{\bar{K}} \longrightarrow \mathbb{R}_{\geq 0}\) such that \(\gamma(v)\) depends only on the restriction \(v|_K\) and the set \(\{ v|_K \, | \, \gamma(v) \neq 0 \}\) is finite. The notation \(O(\mathcal{M}_K)\) will be used to denote a relation that holds up to an \(\mathcal{M}_K\)-constant. For instance \(f \leq g +O(h) + O(\mathcal{M}_K)\) means that there exist a \(C>0\) and an \(\mathcal{M}_K\)-constant \(\gamma\) such that \(f \leq g + C|h| + \gamma\).
Definition: (Local height functions) Let \(K\) be a field with a set of absolute values \(\mathcal{M}_K\). Let \(V\) be a projective variety (not necessarily irreducible) over \(K\) and \(X \subset V\) a closed subscheme. A local height is a function \(\lambda_X \colon V(\bar{K}) \times \mathcal{M}_K \longrightarrow \mathbb{R} \cup \infty\) determined by the following properties:
\begin{enumerate}
\item[(1)] If \(D\) is an effective divisor, we get the usual local height, i.e., \(\lambda_X = \lambda_D\).
\item[(2)] if \(X,X'\) are subschemes \(\lambda_{X \cap X'} = \min{(\lambda_X,\lambda_{X'})}\) (where \(X \cap X'\) denotes the subscheme with ideal sheaf .\(\mathcal{I}_X+\mathcal{I}_{X'}\) ).
\end{enumerate}
and having also many other nice properties:
\begin{enumerate}
\item[(3)] (functoriality) If \(\varphi \colon V \longrightarrow W\) denotes a morphism of varieties and \(X\) is a closed subscheme of \(W\), we have the equality: \[\lambda_{\varphi^{-1} W,V} = \lambda_{X,W}.\]
\item[(4)] Local height functions are bounded below, so up a \(\mathcal{M}_K\)-constant, we can assume that \(\lambda_X \geq 0\).
\end{enumerate}
Remark: Using the local height function associated to the boundary divisor, the local height function machinery can be extended to quasi-projective varieties.
Definition: Let \(\Delta(V)\) denotes the diagonal subvariety in \(V \times V\). The arithmetic distance function on \(V\) is the local height \[\delta_V = \lambda_{\Delta(V)}.\] It is well-defined up to an \(\mathcal{M}_K\)-bounded function and satisfies many nice properties as well, for example:
\begin{enumerate}
\item[(1)] \(\delta(P,R) \geq \min(\delta(P,Q),\delta(Q,R))\)
\item[(2)] \(\lambda_X(Q) \geq \min(\lambda_X(P),\delta(P,Q))\)
\end{enumerate}
Defintion: Let \(\varphi \colon W \longrightarrow V\) be a finite flat morphism between schemes of finite type over a field \(k\). Let \(k'\) be an algebraically closed field containing \(k\). For \(x \in W(k')\), define the multiplicity of \(\varphi\) at \(x\) by the formula \[e_\varphi(x)= \text{length}_{\mathcal{O}_{W_{k'},x}} \mathcal{O}_{W_{k'},x}/\varphi^{-1} m_{\varphi(x)} \mathcal{O}_{W_{k'},x}.\]
For finite flat morphisms \(W \longrightarrow V\), the distribution inequality bounds the arithmetic distance in the target variety in terms of the arithmetic distance of the pre-images in \(W\). In good cases, it will give a distribution relation.
Theorem: (Arithmetic distribution relation/inequality) Let \(\varphi \colon W \longrightarrow V\) be a generically étale finite flat morphism between quasi-projective geometrically integral varieties over \(K\). For all \((P,q,v) \in W(\bar{K}) \times V(\bar{K}) \times v\), we have \[\delta_V(\varphi(P),q,v) \leq \sum_{Q \in W(\bar{K}),\, \varphi(Q)=q} e_\varphi(Q) \delta_W(P,Q,v)+ O(\lambda_{ \partial(V \times W)}(P,q,v))+ O(\mathcal{M}_K).\] We say that we have an arithmetic distribution relation when the above inequality becomes an equality. For example, assuming that \(V,W\) are both smooth, we have an arithmetic relation in the following two situations:
\begin{enumerate}
\item[(1)] The map \(\varphi \colon W \longrightarrow V\) is étale (where there is not ramification).
\item[(2)] The dimensions \(\dim(V)=\dim(W)=1\) (where the ramification divisor is at most zero-dimensional).
\end{enumerate}
Remark: The above inequality does not always became an arithmetic relation. Even when we take a Galois cover \(\varphi \colon W \longrightarrow V\) with Galois group \(\text{Gal}(V/W)=\{\tau_1,\dots,\tau_n\}\), we have an inclusion of associated sheaves of ideals \[\mathcal{I}(\sum_{i=1}^n (1\times \tau_i)^* \Delta(W)) \subset \mathcal{I}((\varphi \times \varphi)^*(\Delta(V))\] that does not always gives an equality of closed schemes. Take for example \(\varphi \colon \mathbb{P}^1 \times \mathbb{P}^1 \longrightarrow \mathbb{P}^1 \times \mathbb{P}^1 \) defined by \(\varphi([x,y],[z,w])=([x^2,y^2],[z,w])\).
Remark: The above inequality can be used to obtain a quantitative inverse theorem, namely, given a finite map, how far apart from the ramification locus and the boundary we need to be, to be able to define a local inverse. Also, the inverse obtained can be shown to behave nicely with respect to the distance functions. In some sense, the distance between points is close to the distance between the inverses.
Theorem: (Inverse function theorem version \(1\)) Suppose that \(V\) and \(W\) are quasi-projective geometrically integral varieties defined over \(K\). Assume that the map \(\varphi \colon W \longrightarrow V\) is a generically étale finite flat surjective morphism of degree \(d\) also defined over K. Let us denote by \(\text{Ann}(\Omega_{W/V})\), the annihilator ideal sheaf of \(\Omega_{W/V}\) and by \(A(\varphi) \subset W\) the closed subscheme defined by \(\text{Ann}(\Omega_{W/V})\).
\begin{enumerate}
\item[(a)] There exist constants \(C_2,C_3\) and \(\mathcal{M}_K\)-constants \(C_4,C_5\) such that the following holds:\\
If the triple \((P,q,v) \in W(K) \times V(K) \times M(K)\) satisfies \[\delta_V (\varphi(P),q;v) \geq d \lambda_{A(\varphi)}(P;v) + C_2 \lambda_{ \partial_{W \times V} }(P,q;v) + C_4(v)\] then there exists a point \(Q \in W(K)\) satisfying \(\varphi(Q) = q\) and \[\delta_W(P,Q;v) \geq \delta_V (\varphi(P),q;v) -(d-1)\lambda_{A(\varphi)}(P;v)- C_3 \lambda_{\partial(W \times V )}(P, q; v) - C_5(v).\]
\item[(b)]If we take \(C_4\) to be an appropriate positive real number, instead of an \(\mathcal{M}_K\) constant, and if we also assume that \(P \notin A(\varphi)\), then the point \(Q\) in (a) is unique.
\end{enumerate}
Remark: The arithmetic distribution relation and the inverse function theorem have been used to study integral points in the following situations:
\begin{enumerate}
\item[(1)] To find uniform height estimates while working with the étale map \([n] \colon A \longrightarrow A\) on an Abelian scheme \(A \longrightarrow T\) over a base variety \(T\).
\item[(2)] To find an analogous of Siegel's theorem while working with Iterates \(f^n\) of a rational map \(f \colon \mathbb{P}^1 \longrightarrow \mathbb{P}^1\) of degree at least two.
\end{enumerate}
The first version (version \(1\)) of the inverse function theorem works simultaneously over several places. In the next version the authors present a stronger result working only over a complete field \(K\). By working over a complete field, the exponents or coefficients of the inverse function theorem are improved from \((d,d-1)\) to \((2,1)\).
Theorem: (Inverse function theorem version \(2\)). Let \( (K, | . |)\) be a complete field. Let \(W, V\) be smooth quasi-projective varieties over \(K\), and let \(\varphi \colon W \longrightarrow V\) be a generically finite generically étale morphism. Let \(E \subset W\) be the closed subscheme defined by the \(0\)-th fitting ideal sheaf of \(\Omega_{W/V}\). Fix arithmetic distance functions \(\delta_W\), \(\delta_V\), a local height function \(\lambda_E\), and a boundary function \(\lambda_{\partial V}\). Let \(B \subset W(K)\) be a bounded subset. Then there are constants \(C_{36}, C_{37}, C_{38}, C_{39} > 0\) and a bounded subset \(\tilde{B} \subset W (K)\) containing \(B\) such that for all \(P \in B\) and \(q \in V(K)\) satisfying \[P \notin E \quad \text{and} \quad \delta_V(\varphi(P),q) \geq 2 \lambda_E(P) + C_{36} \lambda_{\partial V}(q) + C_{37},\] there is a unique \(Q \in \tilde{B}\) satisfying \[\varphi(Q)=q \quad \text{and} \quad \delta_W(P,Q) \geq \delta_V(\varphi(P),q) - \lambda_E(P)-C_{38} \lambda_{\partial V}(q) - C_{39}.\]
Remark: Both versions of the inverse function theorem are suitable to prove results analogous to the continuity of the roots for polynomials. For example from the second version, we could get the following result. Let \( (K, | . |)\) be a complete field. Let \(D \in \mathbb{R}_{>0}\) and \(n \in \mathbb{Z}_{>0}\). Then there are positive constants \(C_{40},C_{41} > 0\) such that the following holds. Suppose that:
\begin{itemize}
\item \(f,g \in K[t]\) are monic polynomials of degree \(n\);
\item The Gauss norms \(|f | \leq D\) and \(|g| \leq D\);
\item There is an \(\alpha \in K\) such that \(f(\alpha) = 0\) and \(|f-g| \leq C_{40} |f''\alpha)|\).
\end{itemize}
Then there is \(\beta \in K\) such that \[g(\beta) = 0 \quad \text{and} \quad |\alpha - \beta||f'(\alpha)| \leq C_{41} |f - g|.\]
Remark: The second version of the inverse function theorem is based on a higher dimensional version of Newton's method. The point \(Q\), will be obtained as limit of a Cauchy sequence of points \(Q_0=P, Q_1, Q_2 \dots\).Elliptic and abelian period spaceshttps://zbmath.org/1472.112122021-11-25T18:46:10.358925Z"Wüstholz, Gisbert"https://zbmath.org/authors/?q=ai:wustholz.gisbertIn his book on transcendental numbers [\textit{T. Schneider}, Einführung in die transzendenten Zahlen Berlin-Göttingen-Heidelberg: Springer-Verlag (1957; Zbl 0077.04703)], Th.~Schneider proposes eight open problems, the third of which is: Try to find transcendence results on elliptic integrals of the third kind. This problem gave rise to a number of papers. A survey on this question is the appendix by the reviewer: \emph{Third kind elliptic integrals and transcendance} to the paper [\textit{C. Bertolin}, J. Pure Appl. Algebra 224, No. 10, Article ID 106396, 27 p. (2020; Zbl 1450.11077)]. The first results were obtained thanks to the appendix by J-P.~Serre \emph{Quelques propriétés des groupes algébriques commutatifs} to the volume [\textit{M. Waldschmidt}, Nombres transcendants et groupes algébriques. (Transcendental numbers and algebraic groups). Complété par deux appendices de Daniel Bertrand et Jean-Pierre Serre. 2e éd. Société Mathématique de France (SMF), Paris (1987; Zbl 0621.10022)]. Among many contributions to Schneider's third problem are the following ones: [\textit{M. Laurent}, C. R. Acad. Sci., Paris, Sér. A 288, 699--701 (1979; Zbl 0402.10038); in: Semin. Delange-Pisot-Poitou, 20e Annee 1978/79, Theorie des nombres, Fasc. 1, Exp. 13, 4 p. (1980; Zbl 0426.10033); J. Reine Angew. Math. 316, 122--139 (1980; Zbl 0419.10034); \textit{E. Reyssat}, Ann. Fac. Sci. Toulouse, Math. (5) 2, 79--91 (1980; Zbl 0439.10021); C. R. Acad. Sci., Paris, Sér. A 290, 439--441 (1980; Zbl 0426.10037); \textit{M. Laurent}, J. Reine Angew. Math. 333, 144--161 (1982; Zbl 0475.10031); \textit{E. Reyssat}, Acta Arith. 41, 291--310 (1982; Zbl 0491.10026); \textit{D. M. Caveny} and \textit{R. Tubbs}, Proc. Am. Math. Soc. 138, No. 8, 2745--2754 (2010; Zbl 1262.11076)]. \par The paper under review provides much more general results on the question of linear independence over the field of algebraic numbers of elliptic integrals. Consider an extension by the multiplicative group of an elliptic curve \(E\), defined by a point \(P\) on the dual of \(E\). Everything is supposed to be defined over the field \({\overline{\mathbb{Q}}}\) of algebraic numbers. Then the dimension of the \({\overline{\mathbb{Q}}}\)--space spanned by \(1\) and the entries of the period matrix is \(8\) if \(E\) has no complex multiplication and \(P\) has infinite order, \(6\) if \(E\) has no complex multiplication and \(P\) has finite order and also if \(E\) has complex multiplication and \(P\) has infinite order, and \(4\) if \(E\) has complex multiplication and \(P\) has finite order. An example from elliptic billiards is given. Some results on abelian hyperelliptic integrals are also given. The main tools are the connections between integrals of the third kind and extensions of elliptic curves and abelian varieties by the multiplicative group (Serre, op. cit., and also [\textit{J.-P. Serre}, Algebraic groups and class fields. Transl. of the French edition. New York etc.: Springer-Verlag (1988; Zbl 0703.14001)]), as well as the author's Analytic Subgroup Theorem [\textit{G. Wüstholz}, Ann. Math. (2) 129, No. 3, 501--517 (1989; Zbl 0675.10025)].The distribution of the maximum of partial sums of Kloosterman sums and other trace functionshttps://zbmath.org/1472.112172021-11-25T18:46:10.358925Z"Autissier, Pascal"https://zbmath.org/authors/?q=ai:autissier.pascal"Bonolis, Dante"https://zbmath.org/authors/?q=ai:bonolis.dante"Lamzouri, Youness"https://zbmath.org/authors/?q=ai:lamzouri.younessLet \(\mathcal{F}=\{\varphi_a\}_{a\in\Omega_m}\) be a family of periodic functions, where \(\Omega_m\) is a non-empty finite set, and for each \(a\in\Omega_m\), \(\varphi_a:\mathbb{Z}\to\mathbb{C}\) is \(m\)-periodic and its Fourier transform \(\widehat{\varphi_a}\) is real-valued and uniformly bounded. For a positive real number \(V\), distribution of the maximum of partial sums of families of \(m\)-periodic complex-valued functions is defined by
\[
\Phi_{\mathcal{F}}(V)=\frac{1}{\#\Omega_m}\,\#\left\{a\in\Omega_m : \frac{1}{\sqrt{m}}\max_{x<m}\left|\sum_{0\leq n\leq x}\varphi_a(n)\right|>V\right\}.
\]
In the paper under review, assuming certain conditions, the authors prove that there exists a constant \(B\) such that for all real numbers \(B\leq V\leq (N/\pi)(\log \log m - 2 \log \log \log m) - B\) one has
\[
\Phi_{\mathcal{F}}(V)=\exp\left(-\exp\left(\frac{\pi}{N}V+O(1)\right)\right).
\]
This general estimate covers some previously known results on the partial sums of character sums, Kloosterman sums and other families of \(\ell\)-adic trace functions.New bounds for exponential sums with a non-degenerate phase polynomialhttps://zbmath.org/1472.112212021-11-25T18:46:10.358925Z"Castryck, Wouter"https://zbmath.org/authors/?q=ai:castryck.wouter"Nguyen, Kien Huu"https://zbmath.org/authors/?q=ai:nguyen.kien-huuSummary: We prove a recent conjecture due to \textit{R. Cluckers} and \textit{W. Veys} [Am. J. Math. 138, No. 1, 61--80 (2016; Zbl 1341.11048)] on exponential sums modulo \(p^m\) for \(m\geq 2\) in the special case where the phase polynomial \(f\) is sufficiently non-degenerate with respect to its Newton polyhedron at the origin. Our main auxiliary result is an improved bound for certain related exponential sums over finite fields. This bound can also be used to settle a conjecture of \textit{J. Denef} and \textit{K. Hoornaert} [J. Number Theory 89, No. 1, 31--64 (2001; Zbl 0994.11038)]on the candidate-leading Taylor coefficient of Igusa's local zeta function associated with a non-degenerate polynomial, at its largest non-trivial real candidate pole.The completed finite period map and Galois theory of supercongruenceshttps://zbmath.org/1472.112382021-11-25T18:46:10.358925Z"Rosen, Julian"https://zbmath.org/authors/?q=ai:rosen.julianSummary: A period is a complex number arising as the integral of a rational function with algebraic number coefficients over a region cut out by finitely many inequalities between polynomials with rational coefficients. Although periods are typically transcendental numbers, there is a conjectural Galois theory of periods coming from the theory of motives. This paper formalizes an analogy between a class of periods called multiple zeta values and congruences for rational numbers modulo prime powers (called supercongruences). We construct an analog of the motivic period map in the setting of supercongruences and use it to define a Galois theory of supercongruences. We describe an algorithm using our period map to find and prove supercongruences, and we provide software implementing the algorithm.Higher moments of arithmetic functions in short intervals: a geometric perspectivehttps://zbmath.org/1472.112552021-11-25T18:46:10.358925Z"Hast, Daniel Rayor"https://zbmath.org/authors/?q=ai:hast.daniel-rayor"Matei, Vlad"https://zbmath.org/authors/?q=ai:matei.vladSummary: We study the geometry associated to the distribution of certain arithmetic functions, including the von Mangoldt function and the Möbius function, in short intervals of polynomials over a finite field \(\mathbb{F}_q\). Using the Grothendieck-Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the \(\ell\)-adic cohomology of these varieties, corresponding to an asymptotic bound on each moment for fixed degree \(n\) in the limit as \(q \to \infty \). The results of this paper can be viewed as a geometric explanation for asymptotic results that can be proved using analytic number theory over function fields.Jacobians of hyperelliptic curves over \(\mathbb{Z}_{n}\) and factorization of \(n\)https://zbmath.org/1472.113102021-11-25T18:46:10.358925Z"Dryło, Robert"https://zbmath.org/authors/?q=ai:drylo.robert"Pomykała, Jacek"https://zbmath.org/authors/?q=ai:pomykala.jacek-mSummary: E. Bach showed that factorization of an integer \(n\) can be reduced in probabilistic polynomial time to the problem of computing exponents of elements in \(\mathbb{Z}_n^\ast\) (in particular the group order of \(\mathbb{Z}_n^\ast\)). It is also known that factorization of square-free integer \(n\) can be reduced to the problem of computing the group order of an elliptic curve \(E/\mathbb{Z}_n\). In this paper we describe the analogous reduction for computing the orders of Jacobians over \(\mathbb{Z}_n\) of hyperelliptic curves Cover \(\mathbb{Z}_n\) using the Mumford representation of divisor classes and Cantor's algorithm for addition. These reductions are based on the group structure of the Jacobian. We also propose other reduction of factorization to the problem of determining the number of points \(|C(\mathbb{Z}_n)|\), which makes use of elementary properties of twists of hyperelliptic curves.Counting points on superelliptic curves in average polynomial timehttps://zbmath.org/1472.113222021-11-25T18:46:10.358925Z"Sutherland, Andrew V."https://zbmath.org/authors/?q=ai:sutherland.andrew-vSummary: We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over \(\mathbb{Q}\) that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curve \(y^m= f(x)\) with \(m\ge 2\) and \(f\in\mathbb{Z}[x]\) any squarefree polynomial of degree \(d\ge 3\), along with a positive integer \(N\). It can compute \(\#X(\mathbb{F}_p)\) for all \(p\le N\) not dividing \(m\text{lc}(f)\text{disc}(f)\) in time \(O(md^3N\log^3N\log\log N)\). It achieves this by computing the trace of the Cartier-Manin matrix of reductions of \(X\). We can also compute the Cartier-Manin matrix itself, which determines the \(p\)-rank of the Jacobian of \(X\) and the numerator of its zeta function modulo \(p\).
For the entire collection see [Zbl 1452.11005].Of limit key polynomialshttps://zbmath.org/1472.130072021-11-25T18:46:10.358925Z"Alberich-Carramiñana, Maria"https://zbmath.org/authors/?q=ai:alberich-carraminana.maria"Boix, Alberto F. F."https://zbmath.org/authors/?q=ai:boix.alberto-f"Fernández, Julio"https://zbmath.org/authors/?q=ai:fernandez.julio"Guàrdia, Jordi"https://zbmath.org/authors/?q=ai:guardia.jordi"Nart, Enric"https://zbmath.org/authors/?q=ai:nart.enric"Roé, Joaquim"https://zbmath.org/authors/?q=ai:roe.joaquimLet \(K\) be a field and \(v\) a valuation on the polynomial ring \(K[x]\), with value group \(\Gamma_v\). For each \(\gamma\in \Gamma_v\), we have the following abelian groups \(\mathcal{P}_{\gamma}^+=\{ g\in K[x]; \mu(g)>\gamma\}\subset\mathcal{P}_{\gamma}=\{ g\in K[x]; \mu(g)\geq\gamma\}\). The graded algebra \(gr_v(K[x])=\oplus_{\gamma\in\Gamma_v}\mathcal{P}_{\gamma}/ \mathcal{P}_{\gamma}^+\) is an integral domain. A MacLane-Vaquie (MLV) key polynomial for \(v\) is a monic polynomial \(\phi\in K[X]\) whose initial term generates a prime ideal in \(gr_v(K[x])\), which cannot be generated by the initial term of a polynomial of smaller degree. The abstract key polynomials for \(v\) are defined in a technical way. In the paper under review, the authors try to find relations between the MLV key polynomials for valuations \(\mu\leq v\) and the abstract key polynomials for \(v\).Generalized \(F\)-signatures of Hibi ringshttps://zbmath.org/1472.130102021-11-25T18:46:10.358925Z"Higashitani, Akihiro"https://zbmath.org/authors/?q=ai:higashitani.akihiro"Nakajima, Yusuke"https://zbmath.org/authors/?q=ai:nakajima.yusukeLet \(R\) be a \(d\)-dimensional Noetherian ring of characteristic \(p>0\). \(R\) is said to have FFRT (finite \(F\)-representation type) if there is a finite set of isomorphism classes of finitely generated indecomposable modules \(\{M_0, \ldots, M_n\}\) such that for any \(e \in \mathbb{N}\) there are \(c_{i,e} \geq 0\), such that \[R^{1/p^e} \cong M_0^{\oplus c_{0,e}}\oplus M_1^{\oplus c_{1,e}}\oplus \cdots \oplus M_n^{\oplus c_{n,e}}.\] The generalized \(F\)-signature of \(M_i\) with respect to \(R\) is \(s(M_i,R):=\underset{e \rightarrow \infty}\lim\displaystyle\frac{c_{i,e}}{p^{ed}}\).
A Hibi ring is a special type of toric ring defined via a poset. For toric rings \(R\) of characteristic \(p\), it is known that \(R\) has FFRT and the indecomposable modules of \(R\) are the conical divisors of \(R\). The goal of this nice paper is to determine the generalized \(F\)-signatures for the conical divisors of a Hibi ring.
The main theorem determines the generalized \(F\)-signature for a conical divisor of a Segre product of polynomial rings of dimension \(d\), which is a Hibi ring, in terms of the number of elements of the symmetric group on a set of \(d\) elements which certain descent properties. The authors claim that the methods used to prove this result can also be used to determine the generalized \(F\)-signature for a conical divisor for other Hibi rings; their running example of a Hibi ring which is not a Segre product provides an illustration of this claim.Monomial generators of complete planar idealshttps://zbmath.org/1472.130162021-11-25T18:46:10.358925Z"Alberich-Carramiñana, Maria"https://zbmath.org/authors/?q=ai:alberich-carraminana.maria"Àlvarez Montaner, Josep"https://zbmath.org/authors/?q=ai:alvarez-montaner.josep"Blanco, Guillem"https://zbmath.org/authors/?q=ai:blanco.guillemLet \((X,O)\) be a germ of smooth complex surface and \(\mathcal{O}_{X,O}\) the ring of germs of holomorphic functions in a neighbourhood of \(O\), and let \(\mathfrak{m}\) be the maximal ideal at \(O\). Let \(\pi:X'\rightarrow X\) be a proper birational morphism that can be achieved as a sequence of blow-ups along a set of points. Given an effective \(\mathbb{Z}\)-divisor \(D\) in \(X'\) we may consider its associated ideal \(\pi_*\mathcal{O}_{X'}(-D)\), whose stalk at \(O\) is denoted as \(H_D\). This type of ideals are complete ideals of \(\mathcal{O}_{X,O}\). Among the class of divisors defining the same complete ideal, we may find a unique maximal representative, which has the property of being antinef. Zariski showed that the above correspondence is an isomorphism of semigroups between the set of complete \(\mathfrak{m}\)-primary ideals and the set of antinef divisors with exceptional support.
In the present paper the authors make this correspondence explicit computationally: given a proper birational morphism \(\pi:X'\rightarrow X\) and an antinef divisor \(D\) in \(X'\), they provide an algorithm that gives a system of generators of the ideal \(H_D\). This algorithm also captures the topological type of \(D\).
Applying the algorithm, the authors provide a method to compute the integral closure of any ideal \(\mathfrak{a}\subseteq \mathcal{O}_{X,O}\). They apply these results to planar ideals, multiplier ideals and a familiy of complete ideals described by valuative conditions given by the interesection multiplicity of the elements of \(\mathcal{O}_{X,O}\) with a fixed germ of plane curve.
The algorithms developed in the paper have been implemented in the computer algebra system \verb|Magma|.Symbolic powers of codimension two Cohen-Macaulay idealshttps://zbmath.org/1472.130222021-11-25T18:46:10.358925Z"Cooper, Susan"https://zbmath.org/authors/?q=ai:cooper.susan-marie"Fatabbi, Giuliana"https://zbmath.org/authors/?q=ai:fatabbi.giuliana"Guardo, Elena"https://zbmath.org/authors/?q=ai:guardo.elena"Lorenzini, Anna"https://zbmath.org/authors/?q=ai:lorenzini.anna"Migliore, Juan"https://zbmath.org/authors/?q=ai:migliore.juan-carlos"Nagel, Uwe"https://zbmath.org/authors/?q=ai:nagel.uwe"Seceleanu, Alexandra"https://zbmath.org/authors/?q=ai:seceleanu.alexandra"Szpond, Justyna"https://zbmath.org/authors/?q=ai:szpond.justyna"Tuyl, Adam Van"https://zbmath.org/authors/?q=ai:van-tuyl.adamLet \(X\) be a codimension two arithmetically Cohen-Macaulay scheme in \(\mathbb{P}^n\) and \(I_X\) its defining ideal. The authors consider the problem of equality between the ordinary and symbolic powers of \(I_X\), that is \(I_X^m=I_X^{(m)}\) for all \(m\geq1\). They survey known results about these equality, and they extend some of these results. They give necessary and sufficient conditions for the above equality, in terms of the number of generators of \(I_X\) for the case of codimension three arithmetically Gorenstein schemes that are locally complete intersection. They also consider the importance of the hypothesis in the presented characterization by dropping some of these hypotheses and analyzing what happens. In the end of the paper, they consider arithmetically Cohen-Macaulay set of points in \(\mathbb{P}_1\times\mathbb{P}_1\) and give new short proofs of known results.Explicit Pieri inclusionshttps://zbmath.org/1472.130252021-11-25T18:46:10.358925Z"Hunziker, Markus"https://zbmath.org/authors/?q=ai:hunziker.markus"Miller, John A."https://zbmath.org/authors/?q=ai:miller.john-a"Sepanski, Mark"https://zbmath.org/authors/?q=ai:sepanski.mark-rSummary: By the Pieri rule, the tensor product of an exterior power and a finite-dimensional irreducible representation of a general linear group has a multiplicity-free decomposition. The embeddings of the constituents are called Pieri inclusions and were first studied by \textit{J. Weyman} [Schur functors and resolutions of minors. Brandeis University, Waltham USA (PhD Thesis) (1980)] and described explicitly by \textit{P. J. Olver} [``Differential hyperforms I'', University of Minnesota, Mathematics Report, 82--101 (1980), \url{https://www-users.cse.umn.edu/~olver/a_/hyper.pdf}]. More recently, these maps have appeared in the work of \textit{D. Eisenbud} et al. [Ann. Inst. Fourier 61, No. 3, 905--926 (2011; Zbl 1239.13023)] and of \textit{S. V. Sam} [J. Softw. Algebra Geom. 1, 5--10 (2009; Zbl 1311.13039)] and Weyman to compute pure free resolutions for classical groups.
In this paper, we give a new closed form, non-recursive description of Pieri inclusions. For partitions with a bounded number of distinct parts, the resulting algorithm has polynomial time complexity whereas the previously known algorithm has exponential time complexity.Virtual resolutions of monomial ideals on toric varietieshttps://zbmath.org/1472.130272021-11-25T18:46:10.358925Z"Yang, Jay"https://zbmath.org/authors/?q=ai:yang.jayGiven a smooth toric variety \(X=X(\Sigma)\) and a \(\mathrm{Pic}(X)\)-graded module \(M\), then a free complex \(F\) of graded \(\mathrm{k}[\Sigma]\)-modules is a virtual resolution of \(M\) if the corresponding complex \(\widetilde{F}\) of vector bundles on \(X\) is a resolution of \(\widetilde{M}\). In the paper under review, the author uses cellular resolutions of monomial ideals to prove an analog of Hilbert's syzygy theorem for virtual resolutions of monomial ideals on smooth toric varieties.Connected sums of graded Artinian Gorenstein algebras and Lefschetz propertieshttps://zbmath.org/1472.130412021-11-25T18:46:10.358925Z"Iarrobino, Anthony"https://zbmath.org/authors/?q=ai:iarrobino.anthony-a"McDaniel, Chris"https://zbmath.org/authors/?q=ai:mcdaniel.chris"Seceleanu, Alexandra"https://zbmath.org/authors/?q=ai:seceleanu.alexandraLet \(A\) and \(B\) be graded Artinian Gorenstein (AG) algebras with the same socle degree, \(d\). Let \(T\) be an AG algebra of socle degree \(k<d\). Suppose that there are surjective maps \(\pi_A : A \rightarrow T\) and \(\pi_B: B \rightarrow T\). The connected sum algebra \(A \#_T B\) is a certain quotient of the fibered product \(A \times_T B\). The connected sum of two AG algebras is again an AG algebra. In this paper the authors first give two alternative descriptions of this construction, including a careful study of how it relates to Macaulay-Matlis duality. They also show that if \(A\) and \(B\) are graded AG algebras satisfying the strong Lefschetz property (SLP) then over \(T = \mathbb F\) (the ground field), the connected sum also has the SLP. This is not true for a general choice of \(T\). However, they also show that connected sums do retain the WLP to some extent.Ideals modulo a primehttps://zbmath.org/1472.130432021-11-25T18:46:10.358925Z"Abbott, John"https://zbmath.org/authors/?q=ai:abbott.john-a"Bigatti, Anna Maria"https://zbmath.org/authors/?q=ai:bigatti.anna-maria"Robbiano, Lorenzo"https://zbmath.org/authors/?q=ai:robbiano.lorenzoThe present paper deals with the problem of reducing an ideal modulo \(p\), i.e. relating an ideal \(I\) in the polynomial ring \(\mathbb{Q}[x_1,\dots,x_n]\) to a corresponding ideal in \(\mathbb{F}_p[x_1,\dots,x_n]\) where \(p\) is a prime number.
The authors define a notion of \(\sigma\)-good prime, where \(\sigma\) is a term ordering and relate it to other similar notions in the literature. Furthermore, the paper introduces a new invariant called universal denominator, which is independent of the term ordering and allows to show that all but finitely many primes are good for \(I\) (see Definiton 2.4).
The methods in the paper make it easy to detect bad primes, a key feature in modular methods (Theorem 4.1 and Corollary 4.2).
The paper includes practical applications to modular computations of Gröbner bases and also includes examples of computations using the computer algebra systems \verb|CoCoA| and \verb|SINGULAR|.Saturations of subalgebras, SAGBI bases, and U-invariantshttps://zbmath.org/1472.130442021-11-25T18:46:10.358925Z"Bigatti, Anna Maria"https://zbmath.org/authors/?q=ai:bigatti.anna-maria"Robbiano, Lorenzo"https://zbmath.org/authors/?q=ai:robbiano.lorenzoLet \(R=K[x_1,\dots ,x_n]\) and \(F\) be a (not necessarily finite) subset of \(R\). Then the subalgebra of \(R\) generated by \(F\) is denoted \(K[F]\). Similar to the notion of Grobner bases for ideals of \(R\), we can define the notion of SAGBI Gröbner basis for \(K[F]\) (see e.g. the paper of the second author and \textit{M. Sweedler} [Lect. Notes Math. 1430, 61--87 (1990; Zbl 0725.13013)] which is regarded as a pioneer work).
Let \(S\) be a \(K\)-subalgebra of the polynomial ring \(R\) , and let \(0 \ne g\in S\). We denote the set \(\bigcup_{i=0}^\infty \{ f \in R \ | \ g^i f \in S\}\) by \(S : g^\infty\).
The problem that the authors address in this paper is as follows: Given polynomials \(g_1,\dots, g_r \in R\), let \(S= K[g_1,\dots, g_r]\) and \(0\ne g \in S\). The problem is to compute a set of generators for \(S : g^\infty\). In the first part of the paper, an algorithm has been presented to compute a set of generators for \(S : g^\infty\) which terminates if and only if it is finitely generated.
In the second part of the paper, the authors consider the case that \(S\) is graded. They show that two operations of computing a SAGBI basis for \(S\) and a set of generators for \(S : g^\infty\) commute and this leads to nice algorithms for computing with \(S : g^\infty\).Coisotropic hypersurfaces in Grassmannianshttps://zbmath.org/1472.130462021-11-25T18:46:10.358925Z"Kohn, Kathlén"https://zbmath.org/authors/?q=ai:kohn.kathlenThis paper studies the so-called higher associated hypersurfaces of a projective variety via the notion of coisotropy. For a \(k\)-dimensional projective variety \(X\) in \(\mathbb{P}^n\), the \(i\)-th associated hypersurface of \(X\) consists of (the Zariski closure of) all \((n-k-1+i)\)-dimensional linear spaces in \(\mathbb{P}^n\) that meet \(X\) at a smooth point non-transversely, which is a subvariety of a Grassmannian. Historically, the cases \(i = 0\) and \(i=1\) have been studied as the Chow and Hurwitz form of \(X\), respectively.
A main result of this paper is a new and direct proof of a characterization (due originally to Gel'fand, Kapranov and Zelevinsky) of such hypersurfaces in the Grassmannian. Namely, a hypersurface in the Grassmannian is the associated hypersurface of some (irreducible) projective variety iff it is coisotropic, i.e. every normal space at a smooth point of the hypersurface is spanned by rank 1 homomorphisms. Since the notion of coisotropy does not depend on the underlying projective variety, this provides an intrinsic description of all higher associated hypersurfaces (hence the term coisotropic hypersurfaces).
In addition, many other results on coisotropic hypersurfaces are given: e.g. the coisotropic hypersurfaces of the projective dual of \(X\) are the reverse of those of \(X\), and the degrees of these are precisely the polar degrees of \(X\). It is also shown that hyperdeterminants are precisely the coisotropic hypersurfaces associated to Segre varieties. Finally, equations for the Cayley variety of all coisotropic forms of a given degree are given, inside Grassmannians of lines. The author has also written a Macaulay2 package to explicitly realize computation of coisotropic hypersurfaces.Parabolic semi-orthogonal decompositions and Kummer flat invariants of log schemeshttps://zbmath.org/1472.140012021-11-25T18:46:10.358925Z"Scherotzke, Sarah"https://zbmath.org/authors/?q=ai:scherotzke.sarah"Sibilla, Nicolò"https://zbmath.org/authors/?q=ai:sibilla.nicolo"Talpo, Mattia"https://zbmath.org/authors/?q=ai:talpo.mattiaSummary: We construct semi-orthogonal decompositions on triangulated categories of parabolic sheaves on certain kinds of logarithmic schemes. This provides a categorification of the decomposition theorems in Kummer flat $K$-theory due to \textit{K. Hagihara} [\(K\)-Theory 29, No. 2, 75--99 (2003; Zbl 1038.19002); Doc. Math. 21, 1345--1396 (2016; Zbl 1357.19001)] and \textit{W. Nizioł} [ibid. 13, 505--551 (2008; Zbl 1159.19003)]. Our techniques allow us to generalize Hagihara and Nizioł's results to a much larger class of invariants in addition to $K$-theory, and also to extend them to more general logarithmic stacks.Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate ringshttps://zbmath.org/1472.140022021-11-25T18:46:10.358925Z"Chirvasitu, Alex"https://zbmath.org/authors/?q=ai:chirvasitu.alexandru"Kanda, Ryo"https://zbmath.org/authors/?q=ai:kanda.ryo"Smith, S. Paul"https://zbmath.org/authors/?q=ai:smith.s-paulSummary: The elliptic algebras in the title are connected graded \(\mathbb{C} \)-algebras, denoted \(Q_{n,k}(E,\tau )\), depending on a pair of relatively prime integers \(n>k\ge 1\), an elliptic curve \(E\) and a point \(\tau \in E\). This paper examines a canonical homomorphism from \(Q_{n,k}(E,\tau)\) to the twisted homogeneous coordinate ring \(B(X_{n/k},\sigma',\mathcal{L}'_{n/k})\) on the characteristic variety \(X_{n/k}\) for \(Q_{n,k}(E,\tau)\). When \(X_{n/k}\) is isomorphic to \(E^g\) or the symmetric power \(S^gE\), we show that the homomorphism \(Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma',\mathcal{L}'_{n/k})\) is surjective, the relations for \(B(X_{n/k},\sigma',\mathcal{L}'_{n/k})\) are generated in degrees \(\le 3\) and the noncommutative scheme \(\text{Proj}_{nc}(Q_{n,k}(E,\tau))\) has a closed subvariety that is isomorphic to \(E^g\) or \(S^gE\), respectively. When \(X_{n/k}=E^g\) and \(\tau =0\), the results about \(B(X_{n/k},\sigma',\mathcal{L}'_{n/k})\) show that the morphism \(\Phi_{|\mathcal{L}_{n/k}|}:E^g \to \mathbb{P}^{n-1}\) embeds \(E^g\) as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.Classification of del Pezzo orders with canonical singularitieshttps://zbmath.org/1472.140032021-11-25T18:46:10.358925Z"Nasr, Amir"https://zbmath.org/authors/?q=ai:nasr.amirSummary: We classify del Pezzo non-commutative surfaces that are finite over their centres and have no worse than canonical singularities. Using the minimal model program, we introduce the minimal model of such surfaces. We first classify the minimal models and then give the classification of these surfaces in general. This presents a complementary result and method to the classification of del Pezzo orders over projective surfaces given by \textit{D. Chan} and \textit{R. S. Kulkarni} [Adv. Math. 173, No. 1, 144--177 (2003; Zbl 1051.14005)].Formal moduli problems and formal derived stackshttps://zbmath.org/1472.140042021-11-25T18:46:10.358925Z"Calaque, Damien"https://zbmath.org/authors/?q=ai:calaque.damien"Grivaux, Julien"https://zbmath.org/authors/?q=ai:grivaux.julienSummary: This paper presents a survey on formal moduli problems. It starts with an introduction to pointed formal moduli problems and a sketch of proof of a Theorem (independently proven by \textit{J. P. Pridham} [Adv. Math. 224, No. 3, 772--826 (2010; Zbl 1195.14012)] and \textit{J. Lurie} [``Derived algebraic geometry X: formal moduli problems'', (2011)]) which gives a precise mathematical formulation for Drinfeld's derived deformation theory philosophy. This theorem provides a correspondence between formal moduli problems and differential graded Lie algebras. The second part deals with Lurie's general theory of deformation contexts, which we present in a slightly different way than the original paper, emphasizing the (more symmetric) notion of Koszul duality contexts and morphisms thereof. In the third part, we explain how to apply this machinery to the case of non-split formal moduli problems under a given derived affine scheme; this situation has been dealt with recently by \textit{J. Nuiten} [Adv. Math. 354, Article ID 106750, 63 p. (2019; Zbl 1433.14007)], and requires to replace differential graded Lie algebras with differential graded Lie algebroids. In the last part, we globalize this to the more general setting of formal thickenings of derived stacks, and suggest an alternative approach to results of \textit{D. Gaitsgory} and \textit{N. Rozenblyum} [A study in derived algebraic geometry. Volume I: Correspondences and duality. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1408.14001)].
For the entire collection see [Zbl 1471.14005].Introductory topics in derived algebraic geometryhttps://zbmath.org/1472.140052021-11-25T18:46:10.358925Z"Pantev, Tony"https://zbmath.org/authors/?q=ai:pantev.tony-g"Vezzosi, Gabriele"https://zbmath.org/authors/?q=ai:vezzosi.gabrieleSummary: We give a quick introduction to derived algebraic geometry (DAG) sampling basic constructions and techniques. We discuss affine derived schemes, derived algebraic stacks, and the Artin-Lurie representability theorem. Through the example of deformations of smooth and proper schemes, we explain how DAG sheds light on classical deformation theory. In the last two sections, we introduce differential forms on derived stacks, and then specialize to shifted symplectic forms, giving the main existence theorems proved in \textit{T. Pantev} et al. [Publ. Math., Inst. Hautes Étud. Sci. 117, 271--328 (2013; Zbl 1328.14027)].
For the entire collection see [Zbl 1471.14005].Characteristic classes of affine varieties and Plücker formulas for affine morphismshttps://zbmath.org/1472.140062021-11-25T18:46:10.358925Z"Esterov, Alexander"https://zbmath.org/authors/?q=ai:esterov.alexander-iSummary: An enumerative problem on a variety \(V\) is usually solved by reduction to intersection theory in the cohomology of a compactification of \(V\). However, if the problem is invariant under a ``nice'' group action on \(V\) (so that \(V\) is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of \(V\). We call this limit the affine cohomology of \(V\) and construct affine characteristic classes of subvarieties of a complex torus, taking values in the affine cohomology of the torus.{
}This allows us to make the first steps in computing affine Thom polynomials. Classical Thom polynomials count how many fibers of a generic proper map of a smooth variety have a prescribed collection of singularities and our affine version addresses the same question for generic polynomial maps of affine algebraic varieites. This notion is also motivated by developing an intersection-theoretic approach to tropical correspondence theorems: they can be reduced to the computation of affine Thom polynomials, because the fundamental class of a variety in the affine cohomology is encoded by the tropical fan of this variety.{
}The first concrete answer that we obtain is the affine version of what were, historically speaking, the first three Thom polynomials--the Plücker formulas for the degree and the number of cusps and nodes of a projectively dual curve. This, in particular, characterizes toric varieties whose projective dual is a hypersurface, computes the tropical fan of the variety of double tangent hyperplanes to a toric variety, and describes the Newton polytope of the hypersurface of non-Morse polynomials of a given degree. We also make a conjecture on the general form of affine Thom polynomials; a key ingredient is the \(n\)-ary fan, generalizing the secondary polytope.On the effective freeness of the direct images of pluricanonical bundleshttps://zbmath.org/1472.140072021-11-25T18:46:10.358925Z"Dutta, Yajnaseni"https://zbmath.org/authors/?q=ai:dutta.yajnaseniSummary: We give effective bounds on the number of twists by ample line bundles, for global generations of pushforwards of log-pluricanonical bundles on klt pairs. This gives a partial answer to a conjecture proposed by \textit{M. Popa} and \textit{C. Schnell} [Algebra Number Theory 8, No. 9, 2273--2295 (2014; Zbl 1319.14022)]. We prove two types of statements: first, more in the spirit of the general conjecture, we show generic global generation with the predicted bound when the dimension of the variety is less than or equal to 4 and more generally, with a quadratic Angehrn-Siu type bound. Secondly, assuming that the relative canonical bundle is relatively semi-ample, we make a very precise statement. In particular, when the morphism is smooth, it solves the conjecture with the same bounds, for certain pluricanonical bundles.All secant varieties of the Chow variety are nondefective for cubics and quaternary formshttps://zbmath.org/1472.140082021-11-25T18:46:10.358925Z"Torrance, Douglas A."https://zbmath.org/authors/?q=ai:torrance.douglas-a"Vannieuwenhoven, Nick"https://zbmath.org/authors/?q=ai:vannieuwenhoven.nickLet \(f\in S^d \mathbb C^{n+1}\) be a homogeneous polynomial of degree \(d\) in \(n+1\) variables. The Chow rank of \(f\) is the minimal integer \(s\) such that \(f\) may be written as
\[
f = \ell_{1,1}\cdots \ell_{1,d} + \cdots + \ell_{s,1}\cdots \ell_{s,d},
\]
where the \(\ell_{i,j}\) are linear forms. This is an important instance of an additive decomposition for a tensor. Tensor decompositions are by now a large field with deep geometric and algebraic roots and yet possess a vast number of applications in many contexts such as complexity, information theory, and machine learning among others.
One geometric feature of the subject arises when one asks what is the Chow rank of a generic \(f\in S^d \mathbb C^{n+1}\). Let \(\mathcal{C}_{d,n}\subset \mathbb P^{\binom{n+d}{d}-1}\) be the projective variety parameterizing products of linear forms in \(S^d \mathbb C^{n+1}\). The variety \(\mathcal{C}_{d,n}\) is called the \textit{Chow variety}. Computing the Chow rank of a generic \(f\in S^d \mathbb C^{n+1}\) is equivalent to finding the smallest \(s\) such that \(\sigma_s(\mathcal{C}_{d,n}) = \mathbb P^{\binom{n+d}{d}-1}\), where \(\sigma_s(\mathcal{C}_{d,n})\) is the \(s\)-th secant variety of the Chow variety. The topic of secant varieties is a delightful chapter of classical algebraic geometry that has attracted more attention in the last decades, partly because of its natural role in additive decompositions and applications thereof.
This nice paper is a contribution to determining dimensions of secants of Chow varieties. The main result is that all secant varieties \(\sigma_s(\mathcal{C}_{d,n})\) have expected dimensions for:
\begin{itemize}
\item any \(n\) and \(d=3\),
\item \(n=3\) and any \(d\).
\end{itemize}
The methods are very combinatorial and rely on a lattice construction generalising a method due to Brambilla and Ottaviani. The base cases of the inductions are treated with a computer-assisted proof.Erratum to: ``Totaro's question on zero-cycles on torsors''https://zbmath.org/1472.140092021-11-25T18:46:10.358925Z"Gordon-Sarney, R."https://zbmath.org/authors/?q=ai:gordon-sarney.reed-leon"Suresh, V."https://zbmath.org/authors/?q=ai:suresh.venapallyErratum to the authors' paper [ibid. 167, No. 2, 385--395 (2018; Zbl 1383.14003)].Néron models of intermediate Jacobians associated to moduli spaceshttps://zbmath.org/1472.140102021-11-25T18:46:10.358925Z"Dan, Ananyo"https://zbmath.org/authors/?q=ai:dan.ananyo"Kaur, Inder"https://zbmath.org/authors/?q=ai:kaur.inderGiven a family \(X\) of smooth curves degenerating to a one-nodal curve \(X_0\), one can consider the Gieseker moduli space \(\mathcal{G}_{X_t}(2,\mathcal{L} _t)\) of semistable vector bundles of rank two and a determinant of odd degree which also varies in a family. For every smooth curve \(X_t\) in the family, one can consider the intermediate Jacobian \(J^i(\mathcal{G}_{X_t}(2,\mathcal{L} _t))\), and all these intermediate Jacobians fit together in one analytic family. To extend this at \(t=0\), different Neron models, constructed Clemens, Saito, Schnell, Zucker and Green-Griffiths-Kerr, are available. In this paper, the authors prove that all these Neron models coincide, and, moreover, they give a description of the special fiber and they prove that it is a semi-abelian varieties in some cases. In particular, in the case of the second intermediate Jacobian, the special fiber is isomorphic to the second generalized intermediate Jacobian of \(\mathcal{G}_{X_0}(2,\mathcal{L}_0)\).The integral Hodge conjecture for 3-folds of Kodaira dimension zerohttps://zbmath.org/1472.140112021-11-25T18:46:10.358925Z"Totaro, Burt"https://zbmath.org/authors/?q=ai:totaro.burtThe paper concerns the proof of the integral Hodge conjecture for some special 3-folds of Kodaira dimension zero such that their canonical bundle has nontrivial sections. The result generalizes earlier work of Voisin and Grobowski [\textit{C. Grabowski}, On the integral Hodge conjecture for 3-folds. Duke University (PhD Thesis) (2004)]. They also prove a similar theorem for integral Tate conjecture. The Hodge conjecture is true for all smooth complex projective 3-folds by Lefschetz (1,1) and hard Lefschetz theorem. The integral Hodge conjecture is the stronger statement that, every element of \(H^{2i}(X, \mathbb{Z})\) whose image in \(H^{2i}(X, \mathbb{C})\) is of type \((i,i)\), is the class of an algebraic cycle of codimension \(i\). The Hodge conjecture is an analogous statement for rational cohomology classes. The integral Tate conjecture for a smooth projective variety \(X\) over the separable closure \(F\) of a finitely generated field \(k\) states that, For \(k\) a finitely generated field of definition of \(X\) with separable closure \(F\) and \(l\) a prime invertible in \(k\) every element of \(H^{2j}(X, \mathbb{Z}_l(j))\) fixed by some open subgroup of \(\mathrm{Gal}(F/k)\) is the class of an algebraic cycle over \(F\) with \(\mathbb{Z}_l\) coefficients. The Tate conjecture is the analogous statement over \(\mathbb{Q}_l\).
Section 2 presents different examples of the varieties which the set up for the results of the paper holds. Let \(X\) be a smooth projective 3-fold of Kodaira dimension zero. The first example is the minimal model of \(X\). In this case Proposition 2.1 says either we have \(H^1(X, \mathcal{O})=H^1(Y, \mathcal{O})\) or \(Y\) is smooth. Thus, the integral Hodge conjecture for smooth 3-folds is a birational invariant property [\textit{C. Voisin}, Jpn. J. Math. (3) 2, No. 2, 261--296 (2007; Zbl 1159.14005)]. In order to prove the integral Hodge conjecture for \(X\) we can assume either \(H^1(X, \mathcal{O})=0\) or else \(K_X\) is trivial. The rest follows from the result of \textit{A. Höring} and \textit{C. Voisin} [Pure Appl. Math. Q. 7, No. 4, 1371--1393 (2011; Zbl 1316.14022)] together with Lemma 3.1 in the text. Examples of this are terminal isolated hypersurface singularities \(xy+f(x,y)=0\) and \(X\) any resolution of a terminal quintic 3-fold. The minimal model exists by \textit{J. S. Milne} [Algebraic groups. The theory of group schemes of finite type over a field. Cambridge: Cambridge University Press (2017; Zbl 1390.14004)], that is \(Y\) is a terminal 3-fold with \(K_Y\) nef and birational to \(X\). Terminal varieties are smooth in codimension 2, and so \(Y\) is smooth outside finitely many points. Since \(X\) has Kodaira dimension zero, the Weil divisor class \(K_Y\) is torsion by the aboundance theorem for 3-folds [\textit{Y. Kawamata}, Invent. Math. 108, No. 2, 229--246 (1992; Zbl 0777.14011)]. The assumption implies \(K_Y\) is trivial and thus corresponds to a Cartier divisor.
The terminal condition implies that \(Y\) has only rational singularities [\textit{M. Reid}, Proc. Symp. Pure Math. 46, 345--414 (1987; Zbl 0634.14003)] and its dualizing sheaf is \(K_Y\). By Goresky-Macpherson \(i_*:H_2(S, \mathbb{Z}) \to H_2(S, \mathbb{Z})\) is surjective for any smooth ample divisor \(S\) in \(Y\). Hence \(H^2(Y, \mathbb{Q}) \to H^2(S, \mathbb{Q})\) is injective. It follows that the mixed Hodge structure on \(H^2(Y, \mathbb{Q})\) is pure [\textit{P. Deligne}, in: Proc. int. Congr. Math., Vancouver 1974, Vol. 1, 79--85 (1975; Zbl 0334.14011)] of weight 2. Writing \(H^2(S, \mathbb{Q})=H^2(Y, \mathbb{Q}) \oplus H^2(Y, \mathbb{Q})^{\perp}\) and using the Hodge decomposition together with Serre duality one obtains \(H^0(Y,TY)=H^1(Y, \mathcal{O})\). Using this fact, the author proves that the identity component of the automorphism group of \(Y\) namely \(Aut^0(Y)\) is an Abelian variety of positive dimension and it preserves the singular locus of \(Y\) which has at most dimension zero. The positivity of the dimension of \(Aut^0(Y)\) implies that the singular locus of \(Y\) must be empty.
The next sort of examples are \(X=\frac{S \times E}{G}\) where \(S\) is a K3 surface and \(E\) is an elliptic curve, and \(G=(\mathbb{Z}/2)^2, (\mathbb{Z}/3)^2, (\mathbb{Z}/4)^2, \mathbb{Z}/2 \times \mathbb{Z}/4, \mathbb{Z}/2 \times \mathbb{Z}/6\) acting \(S\). The Abelian group \(G\) acts on \(E\) by translations. Another example is \(X=\frac{S \times E}{G}\) where \(S\) is an Abelian surface.
Section 3 proves two lemmas about the resolution of an isolated rational 3-fold singularity \(Y\). If \(\pi:X \to Y\) is such a resolution with exceptional locus to be SNC divisor \(D\), then Lemma 3.1 asserts that \(H_2(D, \mathbb{Z})\) is generated by the class of algebraic 1-cycles on \(D\). Lemma 3.2 asserts that in case \(\pi\) is log-canonical, then \(H^i(D, \mathcal{O})=0, \forall i >0\). The proof uses Steenbrink type spectral sequences defined on a stratification of the NC divisor \(D\), [\textit{J. H. M. Steenbrink}, Proc. Symp. Pure Math. 40, 513--536 (1983; Zbl 0515.14003); Ann. Inst. Fourier 47, No. 5, 1367--1377 (1997; Zbl 0889.32035)]. Section 4 presents the proof of Integral Hodge conjecture for varieties \(X\) of the form discussed above, i.e with Kodaira dimension zero and \(h^0(X, K_X) >0\). The proof proceeds for the conjecture on algebraic cycles dimensionwise. For 1-cycles by the Lefschetz (1,1)-theorem the conjecture follows. It remains to prove it for codimension 2 cycles on \(X\). Using the birational invariance for the conjecture we consider \(X \to Y\) to be a birational map which is an isomorphism on the smooth locus and the points above the singular locus are SNC. Then choose \(H\) a very ample line bundle on \(Y\) and \(S\) a smooth surface in the linear system \(|H|\), such that \(H_2(S, \mathbb{Z}) \to H_2(Y, \mathbb{Z})\) is surjective. The Hilbert scheme of smooth surfaces in \(Y\) in the homology class of \(S\) is smooth if \(H\) is sufficiently ample. Then one uses the following lemma.
{Lemma 4.2.} Let \(Y\) be a terminal projective complex 3-fold. Denote by \(S_{t_0}\) the point in the Hilbert scheme \(\mathcal{H}\) corresponding to \(t_0\). Let \(H_2(S_{t_0}, \mathbb{Z})_{van}:=\ker(H_2(S_{t_0}, \mathbb{Z}) \to H_2(Y, \mathbb{Z}))\). Identify \(H^2(S_{t_0}, \mathbb{Z})\) with \(H_2(S_{t_0}, \mathbb{Z})\) by Poincaré and let \(C\) be a non-empty open cone in \(H^2(S_{t_0}, \mathbb{R})_{van}\). Suppose there is a contractible open neighborhood \(U\) of \(t_0\) in \(H\) such that every element of \(H^2(S_{t_0}, \mathbb{Z})_{van} \cap C\) becomes a Hodge class on \(S_t\) for some \(t \in U\). Then every element of \(H_2(Y, \mathbb{Z})\) whose image in \(H_2(Y, \mathbb{C})\) is in \(H_{(1,1)}(Y)\) is algebraic.
The remainder of the proof of Theorem 4.1 comes with the analysis of VHS of family of surfaces in a series of results in Section 5. We list these results without proof to give just an idea about them, for the sake of briefness.
\begin{itemize}
\item Let \(Y\) be a terminal projective complex 3-fold, and \(H\) a very ample line bundle on \(Y\) and \(S\) a smooth surface in the linear system \(|H|\). Suppose that there exists an element \(\lambda \in H^1(S, \Omega^1)_{van}\) such that the linear map
\[
\mu_{\lambda}:H^0(S, N_{S/Y}) \to H^2(S, \mathcal{O})_{van}
\]
is surjective. Then there is a non-empty open cone \(C\) in \(H^2(S_{t_0}, \mathbb{R})_{van}\) and a contractible open \(U\) of \(t_0\) in \(\mathcal{H}\) such that every element of \(H^2(S_{t_0}, \mathbb{Z})_{van} \cap C\) become a Hodge class on \(S_t\) for \(t \in U\) [Corollary 5.1].
\item The map \(\mu_{\lambda}\) is surjective if and only if the map \(\tau_1:F^1H_{van}^2 \to H^2(S, \mathbb{C})_{van}\) which is the restriction of the map \(\tau:H_{van}^2 \to H^2(S, \mathbb{C})_{van}\) induced from the Gauss-Manin connection, is a submersion at \(\tilde{\lambda}\), a lift of \(\lambda\) to \(H^2(S_{t_0}, \mathbb{C})_{van}\), [Lemma 5.2].
\item Proposition 5.3 is an analogue of 4.2 when \(Y\) has trivial canonical bundle.
\item Let \(\mu:V \otimes W \to W\) be a symmetric bilinear form, \(q:W^* \to S^2V^*\) its dual and \(\mu_v:V \to W\) the corresponding linear map. Think of \(q\) as a a linear system of quadrics in \(\mathbb{P}(V^*)\). Then the generic quadric in \(Im(q)\) is smooth if the following holds: there is no closed subvariety \(Z \subset \mathbb{P}(V^*)\) contained in the base locus of \(Im(q)\). and satisfying \(\text{rank}(\mu_v) \leq \dim Z, \ \forall v \in Z\), [Lemma 5.4].
\item Let \(Y\) be a Gorenstein projective 3-fold with isolated canonical singularities and \(H\) as above. For each positive integer \(n\), let \(S\) be a generic surface in \(|nH|\) and define \(V, V', \mu\) associated with \(S\) as above and \(c\) any positive constant. Then there exists a constant \(A\) s.t. the sets
\begin{align*}
\Gamma&=\{v \in V |\text{rank}(\mu_v) \leq cn^2\}\\
\Gamma'&=\{v' \in V'|\text{rank}(\mu_{v'}) \leq cn^2\}
\end{align*}
both have dimension bounded by \(A\) independent of \(n\). Here \(V=H^0(S, K_S)_{van},\ V'=H^0(Y, \mathcal{O}(S))/H^0(Y, \mathcal{O})\), [Lemma 5.5].
\item Let \(Y\) be a terminal projective 3-fold with \(K_Y\) trivial, and \(H\) as before, A a positive integer, and \(S \in |nH|\) be general with \(n\) large enough (depending on \(A\)). Set \(V =H^0(S, K_S)_{van}\) and \(\mu_v:V' \to H^1(S, \Omega^1)\). Then the set \(W=\{v \in V|\text{rank}(\mu_v) <A\}\) is equal to zero, [Lemma 5.6].
\item Let \(S \in |nH|\) be a smooth surface and consider the exact sequence of vector bundles
\begin{align*}
0& \to \Omega_S^1(nH) \to \Omega_Y^2(2nH) \to K_S(2nH) \to 0\\
0& \to \mathcal{O}_S \longrightarrow \Omega_Y^1(nH) \longrightarrow \Omega_S^1(nH) \to 0
\end{align*}
and let \(\delta_1, \delta_2\) be the resulting boundary maps on the long exact cohomology sequence. Then the image of \(\delta_2 \circ \delta_1:H^0(S,K_S(2nH)) \to H^2(S,\mathcal{O})\) is \(H^2(S,\mathcal{O})_{van}\) for large enough \(n\) and any \(S\), [Lemma 5.7].
\end{itemize}
Section 6 is devoted to prove the integral Hodge conjecture under the same set up for \(X\). The proof has two parts.
\begin{itemize}
\item Let \(X\) be a smooth projective variety over the separated closure \(k_s\) of a finitely generated field \(k\). For codimension 1 cycles on \(X\) the Tate conjecture implies the integral Tate conjecture [Lemma 6.2].
\item If \(X\) is a smooth projective 3-fold over the algebraic closure \(\bar{k}\) of a finitely generated field \(k\) of \(\mathrm{char}=0\). Then if Tate conjecture holds for codimension 1-cycles on \(X\) and the integral Tate conjecture holds for \(X_{\mathbb{C}}\) for some \(\bar{k} \hookrightarrow \mathbb{C}\), then the integral Hodge conjecture holds for \(X\) over \(\bar{k}\) [Lemma 6.3].
\end{itemize}Nonlinear traceshttps://zbmath.org/1472.140122021-11-25T18:46:10.358925Z"Ben-Zvi, David"https://zbmath.org/authors/?q=ai:ben-zvi.david"Nadler, David"https://zbmath.org/authors/?q=ai:nadler.davidSummary: We combine the theory of traces in homotopical algebra with sheaf theory in derived algebraic geometry to deduce general fixed point and character formulas. The formalism of dimension (or Hochschild homology) of a dualizable object in the context of higher algebra provides a unifying framework for classical notions such as Euler characteristics, Chern characters, and characters of group representations. Moreover, the simple functoriality properties of dimensions clarify celebrated identities and extend them to new contexts. \par We observe that it is advantageous to calculate dimensions, traces and their functoriality directly in the nonlinear geometric setting of correspondence categories, where they are directly identified with (derived versions of) loop spaces, fixed point loci and loop maps, respectively. This results in universal nonlinear versions of Grothendieck-Riemann-Roch theorems, Atiyah-Bott-Lefschetz trace formulas, and Frobenius-Weyl character formulas. We can then linearize by applying sheaf theories, such as the theories of ind-coherent sheaves and \(\mathcal{D}\)-modules constructed by \textit{D. Gaitsgory} and \textit{N. Rozenblyum} [Contemp. Math. 610, 139--251 (2014; Zbl 1316.14006)]. This recovers the familiar classical identities, in families and without any smoothness or transversality assumptions. On the other hand, the formalism also applies to higher categorical settings not captured within a linear framework, such as characters of group actions on categories.
For the entire collection see [Zbl 1471.14005].Hermitian metrics of positive holomorphic sectional curvature on fibrationshttps://zbmath.org/1472.140132021-11-25T18:46:10.358925Z"Chaturvedi, Ananya"https://zbmath.org/authors/?q=ai:chaturvedi.ananya"Heier, Gordon"https://zbmath.org/authors/?q=ai:heier.gordonIn the article under review, the authors adress the construction of Hermitian metrics with positive holomorphic curvature on compact complex manifolds. The ambiant space is actually the total space of a fibration (holomorphic submersion) \(\pi:X\to Y\) and it is rather natural to ask wether the existence of metrics with positive curvature both on \(Y\) and on the fibers of \(\pi\) implies the existence of such a metric on \(X\).
The corresponding question was answered positively by \textit{C.-K. Cheung} [Math. Z. 201, No. 1, 105--119 (1989; Zbl 0648.53037)] in the opposite case of negative curvature. In the positive curvature case, such metrics were constructed by \textit{N. J. Hitchin} [Proc. Symp. Pure Math. 27, Part 2, 65--80 (1975; Zbl 0321.53052)] on Hirzebruch surfaces.
The main result of this article is a positive answer to the above mentioned question. The proof is quite natural although the computations being a bit involved. As explained by the authors, it is not clear if their method can be used either in the semi-positive case or when the map \(\pi\) has singular fibres. Let us make a final remark: the metric cooked up in this article is merely a Hermitian one, even if the data we started with are Kähler.Birational geometry of moduli spaces of stable objects on Enriques surfaceshttps://zbmath.org/1472.140142021-11-25T18:46:10.358925Z"Beckmann, Thorsten"https://zbmath.org/authors/?q=ai:beckmann.thorstenThe moduli space of stable sheaves on a smooth projective surface is an interesting geometric object and has been studied for a long time. By using the notion of Bridgeland stability condition on triangulate category and its wall-crossing behaviour [\textit{T. Bridgeland}, Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)], there are lots of progress in this direction, including the birational geometry of the moduli space when the surface is \(K3\) [\textit{A. Bayer} and \textit{E. Macrì}, Invent. Math. 198, No. 3, 505--590 (2014; Zbl 1308.14011)] or \(\mathbb{P}^2\) [\textit{C. Li, X. Zhao}, Geom. Topol. 23, No. 1, 347--426 (2019; Zbl 1456.14016)].
The paper under review continues the idea for the \textit{generic} Enriques surface.
Let \(Y\) be an Enriques surface and \(\pi: \widetilde{Y} \rightarrow Y\) be the universal covering map by the \(K3\) surface \(\widetilde{Y}\). Assume that \(Y\) is generic, that is \(\mathrm{Pic}(\widetilde{Y})=\pi^*\mathrm{Pic}(Y)\). Let \(v\) be a Mukai vector so that its pullback \(\pi^*(v)\) is primitive. The main Theorem, Theorem 4.5, established the birational equivalence of two moduli spaces \(M^Y_\sigma(v)\) and \(M^Y_\tau(v)\) for two generic stability conditions \(\sigma, \tau \in \mathrm{Stab}^{\dag}(Y) \) with respect to the Mukai vector \(v\).
To prove Theorem 4.5, the author uses two main ideas. The first idea is to use the notion of constant cycle subvariety. A subvariety is called a constant cycle if all its points become rationally equivalent in the ambient variety. By using the result of [\textit{A. Marian, X. Zhao}, Épijournal de Géom. Algébr., EPIGA Journal Profile 4, Article No. 3, 5 p. (2020; Zbl 1442.14035)], the author shows that the image of the morphism \(\pi^*: M^Y_\sigma(v) \rightarrow M^{\widetilde{Y}}_{\widetilde{\sigma}}(\pi^*(v))\) is a constant cycle Lagrangian. Here \(\widetilde{\sigma}\) is the induced Bridgeland stability on \(\widetilde{Y}\) [\textit{E. Macrì} et al., J. Algebr. Geom. 18, No. 4, 605--649 (2009; Zbl 1175.14010)]. The second idea is to show the corresponding birational morphism \(f: M^{\widetilde{Y}}_{\widetilde{\sigma}_+}(\pi^*(v)) \dashrightarrow M^{\widetilde{Y}}_{\widetilde{\sigma}_-}(\pi^*(v))\) on the \(K3\) surface \(\widetilde{Y}\) is \(i^*\)-equivariant. Here \(i^*\in \mathrm{Aut}(\mathrm{D}^{\mathrm{b}}(\widetilde{Y}))\) is the induced involution by the map \(\pi\). The assumption that \(Y\) is generic is used in the proof of \(i^*\)-equivariance. The moduli spaces \(M^Y_{\sigma_\pm}(v)\) can be identified as fixed set of the involution \(i^*\). Since \(f\) is \(i^*\)-equivariant, then its restriction to the constant cycle Lagrangian \(f|_{M^Y_{\sigma_+}(v)}: M^Y_{\sigma_+}(v) \dashrightarrow M^Y_{\sigma_-}(v)\) gives the birational morphism.
As an application of the main Theorem, the author shows (in Theorem 4.7) that for an arbitrary Enriques surface \(Y\) and a primitive Mukai vector \(v\) of odd rank, then \(M^Y_\sigma(v)\) is birational to some Hilbert scheme of points \(\mathrm{Hilb}^n(Y)\).
As another application, the author shows (in Lemma 4.11) that the existence of global Bayer-Macrì map \(\ell: \mathrm{Stab^{\dag}(Y)} \rightarrow \mathrm{NS}(M_\sigma(v))\) [\textit{A. Bayer} and \textit{E. Macrì}, Invent. Math. 198, No. 3, 505--590 (2014; Zbl 1308.14011); \textit{W. Liu}, Kyoto J. Math. 58, No. 3, 595--621 (2018; Zbl 1412.14009)]. Moreover, the author shows the nef and semiample divisors \(\ell_{\sigma_0,\pm}\in \mathrm{NS}(M_{\sigma_{\pm}}(v)\) are big.
\textit{H. Nuer} and \textit{K. Yoshioka} [Adv. Math. 372, Article ID 107283, 118 p. (2020; Zbl 1454.14041)] obtained more general results of birational equivalence of \(M^Y_\sigma(v)\) and \(M^Y_\tau(v)\) by a different method without assumptions that \(Y\) is generic and \(v\) is primitive.Secant planes of a general curve via degenerationshttps://zbmath.org/1472.140152021-11-25T18:46:10.358925Z"Cotterill, Ethan"https://zbmath.org/authors/?q=ai:cotterill.ethan"He, Xiang"https://zbmath.org/authors/?q=ai:he.xiang"Zhang, Naizhen"https://zbmath.org/authors/?q=ai:zhang.naizhenIn this paper, Osserman's theory of limit linear series, as a generalization of that of Eisenbud-Harris for compact type curves, is reviewed. Built on that, two constructions of a moduli space of inclusions of limit linear series are provided. Both spaces agree set-theoretically, but the second one, described as an intersection of determinantal loci of vector bundles helps proving a smoothing theorem for inclusion of limit linear series in special cases. Based on this, explicit formulas for counting inclusion of limit linear series, and equivalently for the number of secant planes to the image of a curve under a map are calculated. The paper ends with an example showing that the moduli of included limit linear series may have a component of unexpectedly large dimension.
Reviewer's remark: In Definition 4.1, the divisor \(D\) seems undefined -- perhaps \(Y\) may have been intended?Angehrn-Siu type effective basepoint freeness for quasi-log canonical pairshttps://zbmath.org/1472.140162021-11-25T18:46:10.358925Z"Liu, Haidong"https://zbmath.org/authors/?q=ai:liu.haidongSummary: We prove Angehrn-Siu type effective freeness and effective point separation for quasi-log canonical pairs. As a natural consequence, we obtain that these two results hold for semi-log canonical pairs. One of the main ingredients of our proof is inversion of adjunction for quasi-log canonical pairs, which is established in this paper.Jet schemes of quasi-ordinary surface singularitieshttps://zbmath.org/1472.140172021-11-25T18:46:10.358925Z"Cobo, Helena"https://zbmath.org/authors/?q=ai:cobo-pablos.helena"Mourtada, Hussein"https://zbmath.org/authors/?q=ai:mourtada.husseinLet \(X\) be a complex analytically irreducible quasi-ordinary (q.o) singularity, defined by \(f\in \mathbb C\{x_1,x_2\}[z]\). It can be parametrized in the form \(x_1=x_1\), \(x_2=x_2\), \(z=\zeta(x_1,x_2)\) with \(\zeta\in \mathbb C\{x_1,x_2\}[z]\). [\textit{Y.-N. Gau}, Mem. Am. Math. Soc. 388, 109--129 (1988; Zbl 0658.14004)] as shown: A finite set of exponents in the support of the series \(\zeta\) -- they are called the characteristic exponents -- are complete invariants of the topological type of the singularity.
In the paper under review, the authors look for invariants for {\em all} types of singularities. They consider the set of jet schemes of \(X\). For \(m\in\mathbb N\), they define a functor \(F_m\colon \mathbb C\text{-Schemes}\to \text {Sets}\) which is representable by a \({\mathbb C}\)-scheme \(X_m\), the \(m\)th jet scheme. There is a canonical projection \(\pi_m \colon X_m\to X\). In section 4 q.o.\ surfaces with one characteristic exponent are considered. The irreducible components of the \(m\)-jet schemes through the singular locus of a such a surface are described in Th.\ 4.14. A graph \(\Gamma\) is constructed which represents the decomposition of \((\pi^{-1}_m(X_{\text{Sing}}))_{\text{red}}\) for every \(m\). The graph \(\Gamma\) is equivalent to the topological type of the singularity. In section 5 these results are generalized to the general case.Minimal model program for log canonical threefolds in positive characteristichttps://zbmath.org/1472.140182021-11-25T18:46:10.358925Z"Hashizume, Kenta"https://zbmath.org/authors/?q=ai:hashizume.kenta"Nakamura, Yusuke"https://zbmath.org/authors/?q=ai:nakamura.yusuke"Tanaka, Hiromu"https://zbmath.org/authors/?q=ai:tanaka.hiromuThe Minimal Model Program has recently been established for klt threefold pairs defined over algebraically closed fields of characteristic \(p>5\) by work of \textit{C. D. Hacon} and \textit{C. Xu} [J. Am. Math. Soc. 28, No. 3, 711--744 (2015; Zbl 1326.14032)], \textit{P. Cascini}, \textit{H. Tanaka} and \textit{C. Xu} [Ann. Sci. Éc. Norm. Supér. (4) 48, No. 5, 1239--1272 (2015; Zbl 1408.14020)], \textit{C. Birkar} and \textit{J. Waldron} (see [Ann. Sci. Éc. Norm. Supér. (4) 49, No. 1, 169--212 (2016; Zbl 1346.14040); Adv. Math. 313, 62--101 (2017; Zbl 1373.14019)]).
Let \(k\) be a perfect field of characteristic \(p>5\) and let \((X, \Delta)\) be a three dimensional log canonical pair, where \(\Delta\) has real coefficients. The authors show that there exists an MMP with scaling for \((X,\Delta)\) that terminates after finitely many steps. Along the way, the authors prove the cone theorem and a version of the basepoint free theorem in this general setting.Bounds for the stalks of perverse sheaves in characteristic \(p\) and a conjecture of Shende and Tsimermanhttps://zbmath.org/1472.140192021-11-25T18:46:10.358925Z"Sawin, Will"https://zbmath.org/authors/?q=ai:sawin.william-fSummary: We prove a characteristic \(p\) analogue of a result of \textit{D. B. Massey} [Duke Math. J. 73, No. 2, 307--369 (1994; Zbl 0799.32033)] which bounds the dimensions of the stalks of a perverse sheaf in terms of certain intersection multiplicities of the characteristic cycle of that sheaf. This uses the construction of the characteristic cycle of a perverse sheaf in characteristic \(p\) by \textit{T. Saito} [Invent. Math. 207, No. 2, 597--695 (2017; Zbl 1437.14016)]. We apply this to prove a conjecture of \textit{V. Shende} and \textit{J. Tsimerman} [Duke Math. J. 166, No. 18, 3461--3504 (2017; Zbl 1426.11064)] on the Betti numbers of the intersections of two translates of theta loci in a hyperelliptic Jacobian. This implies a function field analogue of the Michel-Venkatesh mixing conjecture about the equidistribution of CM points on a product of two modular curves.Abelian arithmetic Chern-Simons theory and arithmetic linking numbershttps://zbmath.org/1472.140202021-11-25T18:46:10.358925Z"Chung, Hee-Joong"https://zbmath.org/authors/?q=ai:chung.hee-joong"Kim, Dohyeong"https://zbmath.org/authors/?q=ai:kim.dohyeong"Kim, Minhyong"https://zbmath.org/authors/?q=ai:kim.minhyong"Pappas, Georgios"https://zbmath.org/authors/?q=ai:pappas.georgios"Park, Jeehoon"https://zbmath.org/authors/?q=ai:park.jeehoon"Yoo, Hwajong"https://zbmath.org/authors/?q=ai:yoo.hwajongSummary: Following the method of Seifert surfaces in knot theory, we define arithmetic linking numbers and height pairings of ideals using arithmetic duality theorems, and compute them in terms of \(n\)-th power residue symbols. This formalism leads to a precise arithmetic analogue of a ``path-integral formula'' for linking numbers.Independence of \(\ell\) for the supports in the decomposition theoremhttps://zbmath.org/1472.140212021-11-25T18:46:10.358925Z"Sun, Shenghao"https://zbmath.org/authors/?q=ai:sun.shenghaoThe author introduced a notion of a perverse compatible system on \(\mathbb{F}_q\)-schemes, which is a variant of a compactible system for perverse \(t\)-structure. Its relations with the classical definition are investigated in Section 2.
The main theorem of this paper is the following: the proper pushforward of a perverse compatible system of a direct sum of shifted semisimple perverse sheaves is also perverse compatible. The key ingredients of the proof are the existence of \(\ell\)-adic companions on smooth schemes[Theorem 2.5] and Deligne's weight theory.
This main theorem gives a relative version of Gabber's result on the independence of \(\ell\) of intersection cohomology. That is, the support of proper pushforward of the \(\ell\)-adic intersection complex is independent of \(\ell\). At the end of this paper, the author remarked a generalization to \(\mathbb{F}_q\)-Artin stacks.Erratum to: ``Slope filtrations''https://zbmath.org/1472.140222021-11-25T18:46:10.358925Z"André, Yves"https://zbmath.org/authors/?q=ai:andre.yvesCorrects Example 1.2.2.(2) and Lemma 1.2.18 in [the author, ibid. 1, No. 1, 1--85 (2009; Zbl 1213.14039)] and notes that Lemma 1.2.8 should be discarded (Proposition 1.4.18, the only place where this lemma is used, is modified accordingly).\(p\)-adic Tate conjectures and abeloid varietieshttps://zbmath.org/1472.140232021-11-25T18:46:10.358925Z"Gregory, Oliver"https://zbmath.org/authors/?q=ai:gregory.oliver"Liedtke, Christian"https://zbmath.org/authors/?q=ai:liedtke.christianThe authors study a conjecture of Raskind's, which may be thought of as an analogue of the Tate conjecture. In order to state it, let us first fix some notations: suppose \(X/K\) is a smooth and proper variety over a local field \(K/\mathbb{Q}_p\). Let \(\bar{K}\) denote an algebraic closure of \(K\), and \(G_K\) the Galois group \(\mathrm{Gal}(\bar{K}/K)\). We first consider codimension one cycles, and consider the cycle class map
\[
c: \mathrm{NS}\otimes_{\mathbb{Z}} \mathbb{Q}_p\rightarrow H^2_{\text{ét}}(X_{\bar{K}}, \mathbb{Q}_p(1))^{G_K},
\]
where \(\mathrm{NS}\) denotes the Néron-Severi group, and the target denotes the Galois invariants in (Tate-twisted) étale cohomology.
For varieties over a finite field or a number field, the map \(c\) is predicted to be an isomorphism by the Tate conjecture. Although formally similar, the current setting of \(X\) over local fields is in fact rather different; here, the map \(c\) is injective but easily shown to be not surjective, and so if one wants an isomorphism it is necessary to impose further conditions on \(X\). This is what Raskind's conjecture aims to do:
Conjecture (Raskind). The map \(c\) is surjective if \(X\) has totally degenerate reduction.
The phrase \textit{totally degenerate reduction} needs to be made precise, but essentially means that \(X\) has bad reduction, and the components of the special fiber (as well as their intersections, and the intersections of those, and so on) have Chow groups as simple as possible. It can be thought of as a maximal unipotent monodromy condition. Raskind's conjecture is in fact more general and deals with cycles of higher codimensions, but we will abusively refer to the above conjecture as Raskind's conjecture in this review.
One key difference between Tate-type conjectures and, for example, the Hodge conjecture to note is that, while the Hodge conjecture would imply that one can pin down the \(\mathbb{Q}\)-vector space of algebraic cycles inside cohomology, the Tate conjecture only allows one to do so after tensoring to \(\mathbb{Q}_{\ell}\). This observation will be relevant in what follows.
The following is one of the main results of this paper
Theorem. There exist abelian surfaces \(B/K\) for which Raskind's conjecture is false.
Let us give an outline of the proof; we will be brief here since this is nicely explained in the Introduction. Using one of Fontaine's equivalence of categories from \(p\)-adic Hodge theory, one can write
\[
H^2_{\text{ét}}(X_{\bar{K}}, \mathbb{Q}_p(1))^{G_K} \cong H \cap \mathrm{Fil}^1_{dR},
\]
where \(H\) is a \(\mathbb{Q}_p\)-vector space (a subspace of \(H^2_{\text{log-cris}}(X)\)), and the intersection is with the Hodge filtration after comparing to de Rham cohomology. On the other hand, \(H\) has a natural \textit{rational structure}, being spanned by algebraic cycles of the special fiber, and Raskind's conjecture amounts to comparing the intersection of this rational structure with \(\mathrm{Fil}^1_{dR}\), and the a priori larger intersection \(H \cap \mathrm{Fil}^1_{dR}.\) Since one has an explicit description of \(H\) and its rational structure, this gives a strategy to prove the Theorem, and the authors explicitly find very clean counterexamples in this way.
The authors also consider the analogue of Raskind's Conjecture for homomorphisms between abelian varieties and find counterexamples using the same strategy as above; they also prove positive results for abelian varieties isogenous to products of Tate elliptic curves. Throughout the paper the authors work in the more general setting of abeloid varieties. The paper under review is very well written, with the ideas clearly presented.A counterexample to an optimistic guess about étale local systemshttps://zbmath.org/1472.140242021-11-25T18:46:10.358925Z"Lawrence, Brian"https://zbmath.org/authors/?q=ai:lawrence.brian"Li, Shizhang"https://zbmath.org/authors/?q=ai:li.shizhangFrom the text: Relative $p$-adic Hodge theory aims at extending known results in $p$-adic Hodge theory to a $p$-adic local system on a rigid variety. Let $X$ be a geometrically connected, quasi-compact rigid
analytic variety over a $p$-adic field $K$ and let $E$ be a $\mathbb Q_p$ -local system on the étale site $X_{\mathrm{et}}$. In [Invent. Math. 207, No. 1, 291--343 (2017; Zbl 1375.14090)], \textit{R. Liu} and \textit{X. Zhu} showed that if at one point $\tilde{x}\in X(\tilde{K})$ the stalk $\mathbb E_{\tilde{X}}$ is de Rham as a $p$-adic Galois representation, then $\mathbb E$ is a de Rham local system; in particular, the stalk of $\mathbb E_{y}$ at any point $y\in X(\tilde{K})$ would be de Rham as well. They noted that the similar statements replacing ``de Rham'' by either ``crystalline'' or ``semistable'' are both wrong. However, inspired by potential semistability of de Rham representations [\textit{L. Berger}, Invent. Math. 148, No. 2, 219--284 (2002; Zbl 1113.14016)], Liu and Zhu ask [loc. cit., Remark 1.4] if a de Rham local system $\mathbb E$ on $X$ would become semistable after pulling the system back to a finite étale cover of $X$, or even after enlarging the ground field $K$ by a finite extension. While the former guess may well be true, in this paper we construct an example illustrating the failure of the latter.On the monodromy theorem for the family of \(p\)-adic differential equationshttps://zbmath.org/1472.140252021-11-25T18:46:10.358925Z"Mebkhout, Zoghman"https://zbmath.org/authors/?q=ai:mebkhout.zoghmanThe author proves a semi-global monodromy theorem for a \(p\)-adic de Rham bundle in the neighbourhood of a generic point of a hypersurface of a smooth scheme over a perfect field with characteristic \(p>0\) in higher dimensions. The result is formulated using the notion of a Frobenius endomorphism \(\sigma\) of a \(p\)-adic field \(L\), i.e. an endomorphism whose restriction to \(\mathbb{Q}_p\) is the identity, which is continuous and such that for each \(a\in \mathcal{O}_L\) (the ring of integers) one has \(|\sigma (a)-a^p|<1\).The étale fundamental groupoid as a 2-terminal costackhttps://zbmath.org/1472.140262021-11-25T18:46:10.358925Z"Pirashvili, Ilia"https://zbmath.org/authors/?q=ai:pirashvili.iliaIn [Georgian Math. J. 22, 563--571 (2015; Zbl 1339.18006)], the author proves that the 2-categorical version of the Seifert-van Kampen theorem holds for the fundamental groupoid of a topological space. As a continuation, the author studies the étale version in the paper under review. More precisely, for a Noetherian scheme \(X\), the assignment \(\Pi_1\) on the site of finite étale covering of \(X\) sends \(Y\) to its étale fundamental groupoid \(\Pi_1(Y)\). The main result is that \(\Pi_1\) is a costack over the finite étale site. Moreover, this costack is the associated costack of the constant trivial pseduofunctor.A de Rham model for complex analytic equivariant elliptic cohomologyhttps://zbmath.org/1472.140272021-11-25T18:46:10.358925Z"Berwick-Evans, Daniel"https://zbmath.org/authors/?q=ai:berwick-evans.daniel"Tripathy, Arnav"https://zbmath.org/authors/?q=ai:tripathy.arnavLet \(G\) be a compact Lie group, not necessarily connected. The goal of the paper is to construct a version of equivariant elliptic cohomology using differential forms. Definition resembles the Cartan model for equivariant cohomology. The output is a sheaf of commutative differential graded algebras over certain stack Bun\(_G(\mathcal E)\) classifying \(G\)-bundles over the universal elliptic curve. If \(G=T\) is a torus of rank \(r\) then Bun\(_T(\mathcal E)\) is the fiber product of \(r\) copies of the dual elliptic universal curves over the moduli space of elliptic curves. For a smooth manifold \(M\) the elliptic cohomology is the sheaf \(\widehat{\mathrm{Ell}}^\bullet_G(M)\) glued from local data containing the following information: for each pair of commuting elements of \(h_1,h_2\in G\) one considers the fixed point set \(M^{\langle h_1,h_2\rangle}\) and the differential forms with values in function \(U\subset\mathrm{Bun}_T(\mathcal E)\) satisfying certain coherence constrains. For a representation \(G\to\mathrm{Spin}(2n)\) the Euler class in this model is a section (a Mathai-Quillen type cocycle) of the sheaf \(\widehat{\mathrm{Ell}}^\bullet_{\mathrm{Spin}(2n)}(pt)\) twisted by the Looijenga line bundle \(\mathcal L_1\). It is essentially given by the Weierstrass sigma function. (Similarly for a representation in \(U(n)\).) It is shown that these classes give a complex analytic equivariant refinement of the MString-orientation in elliptic cohomology.
Although the torsion in the de Rham model is lost, the current definition unifies the construction of \textit{I. Grojnowski} [Lond. Math. Soc. Lect. Note Ser. 342, 114--121 (2007; Zbl 1236.55008)] for connected groups and the approach of \textit{J. A. Devoto} [Mich. Math. J. 43, No. 1, 3--32 (1996; Zbl 0871.55004)] for finite groups.Computation of the divided Frobenius modulo \(p\) on the crystalline cohomology of certain covers of the projective linehttps://zbmath.org/1472.140282021-11-25T18:46:10.358925Z"Pierrot, Amandine"https://zbmath.org/authors/?q=ai:pierrot.amandineSummary: In this paper we describe a family of tamely ramified coverings of the projective line over a finite field, for which we compute the matrix of the divided crystalline Frobenius. The formulas we obtain generalize the classical Hasse-Witt formulas in the case of hyperelliptic curves. Our result relies on a result of Huyghe-Wach which shows that the divided crystalline Frobenius coincides with the explicit morphism, constructed by \textit{P. Deligne} and \textit{L. Illusie} [Invent. Math. 89, 247--270 (1987; Zbl 0632.14017)], for their proof of the degeneration of the Hodge-de Rham spectral sequence in the algebraic case.
For the entire collection see [Zbl 1471.11005].Motivic Mahowald invariants over general base fieldshttps://zbmath.org/1472.140292021-11-25T18:46:10.358925Z"Quigley, J. D."https://zbmath.org/authors/?q=ai:quigley.james-dSummary: The motivic Mahowald invariant was introduced in [\textit{J. D. Quigley}, Algebr. Geom. Topol. 19, No. 5, 2485--2534 (2019; Zbl 1436.55016)] and [\textit{J. D. Quigley}, J. Topol. 14, No. 2, 369--418 (2021; Zbl 07381853)] to study periodicity in the \(\mathbb{C} \)- and \(\mathbb{R} \)-motivic stable stems. In this paper, we define the motivic Mahowald invariant over any field \(F\) of characteristic not two and use it to study periodicity in the \(F\)-motivic stable stems. In particular, we construct lifts of some of Adams' classical \(v_1\)-periodic families [\textit{J. F. Adams}, Topology 5, 21--71 (1966; Zbl 0145.19902)] and identify them as the motivic Mahowald invariants of powers of \(2+\rho \eta \).Rational approximations on toric varietieshttps://zbmath.org/1472.140302021-11-25T18:46:10.358925Z"Huang, Zhizhong"https://zbmath.org/authors/?q=ai:huang.zhizhongIn the article under review, the author is motivated by the rational curve conjectures of \textit{Y. I. Manin} [Compos. Math. 85, No. 1, 37--55 (1993; Zbl 0780.14022)] and \textit{D. McKinnon} [J. Algebr. Geom. 16, No. 2, 257--303 (2007; Zbl 1140.14016)]. The guiding principal is that the best approximations of a general rational point in a rationally connected variety, defined over a number field \(\mathbf{K}\), should be achieved on the subvarieties that are swept out by small degree free rational curves.
The present article builds on the work of \textit{D. McKinnon} and \textit{M. Roth} [Invent. Math. 200, No. 2, 513--583 (2015; Zbl 1337.14023)]. Specifically, the author studies the case of \(\mathbf{K}\)-rational points in split nonsingular toric varieties \(X\). It is required that \(X\) has dimension \(r \geq 2\). Moreover, throughout the article, it is assumed that \(\overline{\operatorname{Eff}}(X)\), the cone of pseudoeffective divisors, is simplicial.
The author's main result gives a description of the best approximation rational curves, with respect to ample, and, more generally, nef line bundles, that pass through a given \(\mathbf{K}\)-rational point \(Q\) of the algebraic torus \(\mathbb{G}_m^r \subseteq X\). In more precise terms, for the case of an ample line bundle \(L\), the author proves that the best approximations for \(Q\) are properly achieved on the subvariety that is swept out by the minimal \(L\)-degree free rational curves through \(Q\).
One aspect to the proof of this result, involves the theory of universal torsors and the description of height functions on toric varieties, which are given in [\textit{P. Salberger}, Astérisque. 251, 91--258 (1998; Zbl 0959.14007)]. Another is an auxiliary result, which is of an independent interest. It gives necessary and sufficient conditions for positive primitive fan relations to correspond to very free rational curves.A note on Higgs-de Rham flows of level zerohttps://zbmath.org/1472.140312021-11-25T18:46:10.358925Z"Sheng, Mao"https://zbmath.org/authors/?q=ai:sheng.mao"Tong, Jilong"https://zbmath.org/authors/?q=ai:tong.jilongSummary: The notion of Higgs-de Rham flows was introduced by \textit{G. Lan} et al. [J. Eur. Math. Soc. (JEMS) 21, No. 10, 3053--3112 (2019; Zbl 1444.14048)], as an analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory. In this paper we investigate a small part of this theory, and study those Higgs-de Rham flows which are of level zero. We improve the original definition of level-zero Higgs-de Rham flows (which works for general levels), and establish a Hitchin-Simpson-type correspondence between such objects and certain representations of fundamental groups in positive characteristic, which generalizes a classical results of \textit{N. Katz} [Lect. Notes Math. 317, 167--200 (1973; Zbl 0259.14007)]. We compare the deformation theories of two sides in the correspondence, and translate the Galois action on the geometric fundamental groups of algebraic varieties defined over finite fields into the Higgs side.\(\mathbb{F}_{p^2}\)-maximal curves with many automorphisms are Galois-covered by the Hermitian curvehttps://zbmath.org/1472.140322021-11-25T18:46:10.358925Z"Bartoli, Daniele"https://zbmath.org/authors/?q=ai:bartoli.daniele"Montanucci, Maria"https://zbmath.org/authors/?q=ai:montanucci.maria"Torres, Fernando"https://zbmath.org/authors/?q=ai:torres.fernando.1Let \({\mathbb{F}} = GF(q^2)\) denote the finite field with \(q^2\) elements. For a divisor \(m\) of \(q+1\), denote by \(H_{m}\) the curve \(y^{m} = x^{q}+x\). In particular, \(H_{q+1}\) denotes the Hermitian curve \(y^{q+1} = x^{q}+x\). It is known, for example, that \(H_{m}\) is Galois-covered by \(H_{q+1}\) over \({\mathbb{F}}\) (see Lemma 2.2 in the paper under review).
A (projective, non-singular, geometrically irreducible) curve \(X\) of genus \(g\) over \({\mathbb{F}}\) is called \({\mathbb{F}}\)-maximal if the number of its \({\mathbb{F}}\)-rational points is maximum possible, i.e., it attains the Hasse-Weil bound, \(|X({\mathbb{F}})=q^2+1+2qg\).
The main result of the authors deals with the case where \(q=p\) is a prime. Their main result is: Let \(X\) be an \({\mathbb{F}}\)-maximal curve with genus \(g\geq 2\). If \(|Aut(X)|>84(g-1)\) then \(X\) is Galois-covered by the Hermitian curve \(H_{q+1}\) over \({\mathbb{F}}\). The proof is broken into the cases \(p\leq 5\) and \(p>5\).
The bound on the size of the automorphism group of \(X\) is sharp. The authors give an example (where \(q=p=71\)) of an \({\mathbb{F}}\)-maximal curve with genus \(g=7\), where \(|Aut(X)|=84(7-1)\), and yet is not Galois-covered by \(H_{72}\).
For further details, the reader is referred to the paper.
The paper is dedicated to the memory of the last-named author, Fernando Torres, who died of Covid-19 in May 2020 while the paper was in press. May his memory be a blessing.Artin-Mazur-Milne duality for fppf cohomologyhttps://zbmath.org/1472.140332021-11-25T18:46:10.358925Z"Demarche, Cyril"https://zbmath.org/authors/?q=ai:demarche.cyril"Harari, David"https://zbmath.org/authors/?q=ai:harari.davidSummary: We provide a complete proof of a duality theorem for the fppf cohomology of either a curve over a finite field or a ring of integers of a number field, which extends the classical Artin-Verdier Theorem in étale cohomology. We also prove some finiteness and vanishing statements.Hurwitz theory of elliptic orbifolds. Ihttps://zbmath.org/1472.140342021-11-25T18:46:10.358925Z"Engel, Philip"https://zbmath.org/authors/?q=ai:engel.philip-miltonSummary: An \textit{elliptic orbifold} is the quotient of an elliptic curve by a finite group. In [Invent. Math. 145, No. 1, 59--103 (2001; Zbl 1019.32014)], \textit{A. Eskin} and \textit{A. Okounkov} proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular forms for \(\operatorname{SL}2_(\mathbb{Z})\). In [Prog. Math. 253, 1--25 (2006; Zbl 1136.14039)], they generalized this theorem to branched covers of the quotient of an elliptic curve by \(\pm 1\), proving quasimodularity for \(\Gamma_0(2)\). We generalize their work to the quotient of an elliptic curve by \(\langle\zeta N\rangle\) for \(N=3, 4, 6\), proving quasimodularity for \(\Gamma(N)\), and extend their work in the case \(N=2\).
It follows that certain generating functions of hexagon, square and triangle tilings of compact surfaces are quasimodular forms. These tilings enumerate lattice points in moduli spaces of flat surfaces. We analyze the asymptotics as the number of tiles goes to infinity, providing an algorithm to compute the Masur-Veech volumes of strata of cubic, quartic, and sextic differentials. We conclude a generalization of the Kontsevich-Zorich conjecture: these volumes are polynomial in \(\pi\).The cuspidalisation of sections of arithmetic fundamental groups. IIhttps://zbmath.org/1472.140352021-11-25T18:46:10.358925Z"Saïdi, Mohamed"https://zbmath.org/authors/?q=ai:saidi.mohamed-redaSummary: In this paper, which is a sequel to [\textit{M. Saïdi}, Adv. Math. 230, No. 4--6, 1931--1954 (2012; Zbl 1260.14036)], we investigate the theory of cuspidalisation of sections of arithmetic fundamental groups of hyperbolic curves to cuspidally \(i\)-th and \(2 / p\)-th step prosolvable arithmetic fundamental groups. As a consequence we exhibit two, necessary and sufficient, conditions for sections of arithmetic fundamental groups of hyperbolic curves over \(p\)-adic local fields to arise from rational points. We also exhibit a class of sections of arithmetic fundamental groups of \(p\)-adic curves which are orthogonal to \(\mathrm{Pic}^\wedge\), and which satisfy (unconditionally) one of the above conditions.Gluing curves of genus 1 and 2 along their 2-torsionhttps://zbmath.org/1472.140362021-11-25T18:46:10.358925Z"Hanselman, Jeroen"https://zbmath.org/authors/?q=ai:hanselman.jeroen"Schiavone, Sam"https://zbmath.org/authors/?q=ai:schiavone.sam"Sijsling, Jeroen"https://zbmath.org/authors/?q=ai:sijsling.jeroenLet \(A\) be an abelian variety over a field \(k\). It is well-known that there is an isogeny decomposition (Poincaré's Complete Reducibility Theorem) in terms of simple abelian subvarieties \[A \sim B_1^{n_1} \times \cdots \times B_r^{n_r}\] that are pairwise non-isogenous over \(k\). This decomposition is unique up to reordering the factors.
In the case that \(A\) equals the Jacobian variety of a curve \(Z\), there exist algorithms to calculate the aforementioned decomposition in terms of the Jacobians of curves over small extensions of \(k\) whenever possible. The decomposition of the Jacobian of curves has been extensively studied.
The article under review consider a different approach to this problem. The authors aim to develop algorithms to construct an abelian variety \(A\) given factors \(B_i\) as before, in some special cases.
Let X be a curve of genus 1 and let \(Y\) be a curve of genus 2, defined over a base field \(k\) whose characteristic is different from 2. The main result of the paper provides criteria for the existence of a curve \(Z\) over \(k\) whose Jacobian is, up to a special isogeny, isogenous to the products of the Jacobians of \(X\) and \(Y\). The authors also developed algorithms to construct the curve \(Z\) once equations for \(X\) and \(Y\) are given.Automorphism group of a moduli space of framed bundles over a curvehttps://zbmath.org/1472.140372021-11-25T18:46:10.358925Z"Alfaya, David"https://zbmath.org/authors/?q=ai:alfaya.david"Biswas, Indranil"https://zbmath.org/authors/?q=ai:biswas.indranilThe authors compute the automorphism group of the moduli space of framed vector bundles on a smooth complex projective curve \(X\). A framed vector bundle is a pair \((E,\alpha)\), consisting of a vector bundle \(E\) of rank \(r\geq 2\) and a nonzero linear map \(\alpha: E_x\rightarrow \mathbb C^r\) from a fiber over a fixed point \(x\in X\) to \(\mathbb C^r\), where the map \(\alpha\) is called a framing. Using the GIT to construct the moduli space of framed vector bundles, the \(\tau\)-stability of a framed vector bundle is introduced by Huybrechts and Lehn, where \(\tau>0\) is a real number. For a given real number \(\tau>0\), a frame bundle \((E,\alpha)\) is \(\tau\)-stable (resp. \(\tau\)-semistable) if for all proper subbundles \(0\subsetneq F\subsetneq E\),
\[
\frac{\mathrm{degree}(F)-\epsilon(F,\alpha)\tau}{\mathrm{rank}(F)}<\frac{\mathrm{degree}(E)-\tau}{\mathrm{rank}(E)}\quad(\text{resp. }\leq)
\]
where
\[
\epsilon(F,\alpha)=\begin{cases}
1&\text{if \(F_x\nsubseteq\mathrm{Ker}(\alpha)\)}\\
0&\text{if \(F_x\subseteq\mathrm{Ker}(\alpha)\)}.
\end{cases}
\]
Fix a line bundle \(\xi\) on \(X\). Let \(\mathcal F=\mathcal F(X,x,r,\xi,\tau)\) be the moduli space of \(\tau\)-semistable framed vector bundles \((E,\alpha)\) on \(X\) with rank \(r\geq 2\) and \(\mathrm{det}(E)\cong\xi\). In fact, it is a complex projective varieties. The stability parameter \(\tau\) is called generic if there do not exist strictly \(\tau\)-semistable framed vector bundles in \(\mathcal F\). If \(\tau\) is generic, the moduli space \(\mathcal F\) is smooth. There is a natural \(\mathrm{PGL}_r(\mathbb C)\)-action on a moduli space of framed vector bundles of rank \(r\). For \([G]\in \mathrm{PGL}_r(\mathbb C)\) with \(G\in \mathrm{GL}_r(\mathbb C)\), the action of \([G]\) on a frame vector bundle \((E,\alpha)\) is described by:
\[
[G]\cdot(E,\alpha)=(E, G\circ\alpha).
\]
The main result of this paper is: For \(\tau\) and \(\tau^\prime\) are generic, if there is an isomorphism \(\Psi: \mathcal F(X,x,r,\xi,\tau)\rightarrow\mathcal F(X^\prime,x^\prime,r^\prime,\xi^\prime,\tau^\prime)\), where the genera of \(X\) and \(X^\prime\) satisfy \(g\geq\max\{2+\tau,4\}\) and \(g^\prime\geq\max\{2+\tau^\prime,4\}\), respectively. Then, \(r=r^\prime\) and there exists an isomorphism \(\sigma: X^\prime\rightarrow X\) such that \(\sigma(x^\prime)=x\), and the isomorphism \(\Psi\) is a combination of the following three types of transformations:
\begin{enumerate}
\item pullback with respect to the isomorphism \(\sigma\);
\item tensorization with a line bundle \(L\) on \(X\);
\item the natural action of \(PGL_r\),
\end{enumerate}
where \(\sigma\) and \(L\) satisfy the relation \(\sigma^*(\xi\otimes L^{\otimes r})\cong\xi^\prime\).A canonical connection on bundles on Riemann surfaces and Quillen connection on the theta bundlehttps://zbmath.org/1472.140382021-11-25T18:46:10.358925Z"Biswas, Indranil"https://zbmath.org/authors/?q=ai:biswas.indranil"Hurtubise, Jacques"https://zbmath.org/authors/?q=ai:hurtubise.jacques-cLet \(X\) be a compact Riemann surface of genus \(g\geq 2\) with canonical line bundle \(K_X\), \(r\) a natural number, and \(\mathcal{M}\) the moduli space of stable vector bundles on \(X\) of rank \(r\) and degree \(0\). Given a theta characteristic \(K_{X}^{1/2}\), \(D_{\Theta}:=\{E\in\mathcal{M}: H^{0}(X,E\otimes K_{X}^{1/2})\neq 0\}\) denotes the theta divisor, and \(\Theta\) denotes the associated theta line bundle over \(\mathcal{M}\).
The authors consider two fibered spaces over \(\mathcal{M}\). On the one hand, the moduli space \(\mathcal{C}\) of holomorphic connections, \(E\rightarrow E\otimes K_X\), with \(E\) a stable vector bundle of rank \(r\). The forgetful map induces a holomorphic map, \(\mathcal{C}\rightarrow \mathcal{M}\). On the other hand, the fiber bundle Conn\((\Theta)\rightarrow \mathcal{M}\) given by the sheaf of holomorphic connections on \(\Theta\). Sections of Conn\((\Theta)\rightarrow \mathcal{M}\) over an open subset \(U\subset \mathcal{M}\) are in one-to-one correspondence with holomorphic connections on \(\Theta|_{U}\). Both \(\mathcal{C}\) and Conn\((\Theta)\) are holomorphic \(T^{*}\mathcal{M}\)-torsors and carry holomorphic symplectic forms, \(\Phi_1\) and \(\Phi_2\), respectively.
Recently, the authors have constructed a \(\mathcal{C}^{\infty}\) isomorphism of torsors \(F:\mathcal{C}\simeq \mathrm{Conn}(\Theta)\) with the property that \(F^{*}\Phi_2 \simeq \Phi_1\) up to multiplication by \(2r\); that is, it preserves the symplectic structures. Further, they have proved that \(F\) is, in fact, holomorphic. The main result of this article is that the restriction of \(F\) to \(\mathcal{M}_0 :=\mathcal{M}\setminus D_{\Theta}\) can be constructed by purely algebraic methods. More precisely, the authors construct holomorphic sections, \(\phi\) and \(\tau\), of the torsors \(\mathcal{C}|_{\mathcal{M}_0}\rightarrow \mathcal{M}_0\) and \(\mathrm{Conn}(\Theta)|_{\mathcal{M}_0}\rightarrow \mathcal{M}_0\) respectively, and prove that the holomorphic map \(G\) defined by \(G(\delta^0 (\phi(E),\nu))=\eta^0(\tau(E),2r\cdot \nu)\) is a holomorphic isomorphism between \(\mathcal{C}|_{\mathcal{M}_0}\) and \(\mathrm{Conn}(\Theta)|_{\mathcal{M}_0}\), and it coincides with \(F|_{\mathcal{M}_0}\). Here, \(\delta^0\) and \(\eta^0\) are the actions of \(T^{*}\mathcal{M}_{0}\) on \(\mathcal{C}|_{\mathcal{M}_0}\) and \(\mathrm{Conn}(\Theta)|_{\mathcal{M}_0}\) respectively.
As a consequence of the main result is that the holomorphic isomorphism \(F\), which preserves the symplectic structures, depends holomorphically on the base Riemann surface. This fact lets the authors to extended the holomorphic isomorphism \(F\) to a relative context.The field of moduli of singular \(K3\) surfaceshttps://zbmath.org/1472.140392021-11-25T18:46:10.358925Z"Laface, Roberto"https://zbmath.org/authors/?q=ai:laface.robertoIt is known that any CM elliptic curves admit the field of moduli which is a Galois extension of the rational field. An analogous argument makes it possible to consider the fields of moduli for singular abelian and \(K3\) surfaces.
For a singular \(K3\) surface \(X\) with its CM field \(K\), let \(G_K\) and \(G_\mathbb{Q}\) respectively subgroups of \({\mathrm {Gal}}(\bar{K}\slash K)\) and of \({\mathrm {Gal}}(\mathbb{C}\slash\mathbb{Q})\) such that the action on \(X\) is invariant in the class of \(X\). Denote by \(M_K\) and \(M_\mathbb{Q}\) respectively the field of \(K\)-moduli and absolute field of moduli, that are by definition subfields of \(\bar{K}\) and of \(\bar{\mathbb{Q}}\) preserved by the actions of \(G_K\) and \(G_\mathbb{Q}\).
Following preliminary studies of the groups \(G_K\) and \(G_\mathbb{Q}\), the author can prove that the absolute field \(M_\mathbb{Q}\) is a Galois extension of the field \(M_K\), and that the fields \(M_K\) and \(M_\mathbb{Q}\) are respectively extensions of \(K\) and of \(\mathbb{Q}\) of degree being the genus of the transcendental lattice of \(X\).
It is concluded that there exist infinitely many singular \(K3\) surfaces with a field of \(K\)-moduli.An example of birationally inequivalent projective symplectic varieties which are D-equivalent and L-equivalenthttps://zbmath.org/1472.140402021-11-25T18:46:10.358925Z"Okawa, Shinnosuke"https://zbmath.org/authors/?q=ai:okawa.shinnosukeThere are many different ways to try to classify algebraic varieties. Among those are
\begin{itemize}
\item birational equivalence: whether a birational map \(X \dashrightarrow Y\) exists;
\item D-equivalence: whether there is an equivalence \(\mathcal D^b(\mathrm{coh}(X)) \cong \mathcal D^b(\mathrm{coh}(Y))\);
\item L-equivalence: whether \(X\) and \(Y\) have the same class in \(K_0(\mathcal{V}ar)[\mathbb L^{-1}]\), where \(\mathbb L = [\mathbb A_1]\).
\end{itemize}
These three attempts are related, but in a mysterious way.
The main result of this article is the following. Given a pair \(X,Y\) of two K3 surfaces of Picard number 1 and degree 2d, which are D- and L-equivalent. Then the Hilbert schemes of points \(X^{[n]}\) and \(Y^{[n]}\) are D- and L-equivalent. If \(n>2\) and if there are integer solutions to the equation
\[
(n-1)x^2 - dy^2 = 1,
\]
then \(X\) and \(Y\) are birationally \emph{in}equivalent.
For the proof, the author uses that a birational morphism \(\phi \colon X^{[n]} \dashrightarrow Y^{[n]}\) induces an isometry of Picard groups, hence preserves the movable cone. The equation of the main result comes now from the description of the movable cone, as obtained in [\textit{A. Bayer} and \textit{E. Macrì}, Invent. Math. 198, No. 3, 505--590 (2014; Zbl 1308.14011)].
To get examples, one can consider a very general K3 surface \(X\) of degree 12 and its Fourier-Mukai partner \(Y\), see [\textit{B. Hassett} and \textit{K.-W. Lai}, Compos. Math. 154, No. 7, 1508--1533 (2018; Zbl 1407.14010)] and [\textit{A. Ito} et al., Sel. Math., New Ser. 26, No. 3, Paper No. 38, 27 p. (2020; Zbl 1467.14051)]. If one chooses \(n=6y^2+2\) for an integer \(y\), then \((1,y)\) is an integer solution of the equation above. Hence, the corresponding hyperkähler varieties \(X^{[n]}\) and \(Y^{[n]}\) give examples of varieties, which are D-equivalent (already known by [\textit{D. Ploog}, Adv. Math. 216, No. 1, 62--74 (2007; Zbl 1167.14031)]) and L-equivalent, but not birationally equivalent.
More results about birational (in)equivalence of such hyperkähler varieties are obtained in [\textit{C. Meachan} et al., Math. Z. 294, No. 3--4, 871--880 (2020; Zbl 1469.14011)] and [\textit{K. Yoshioka}, Math. Ann. 321, No. 4, 817--884 (2001; Zbl 1066.14013)].Rational curves on fibered varietieshttps://zbmath.org/1472.140412021-11-25T18:46:10.358925Z"Anella, Fabrizio"https://zbmath.org/authors/?q=ai:anella.fabrizioLet \(X\) be a complex projective manifold with trivial first Chern class \(c_1(X)\) but non-zero second Chern class \(c_2(X) \neq 0\), e.g. \(X\) is a Calabi-Yau or hyper-Kähler manifold. Then one expects that \(X\) contains a rational curve, this is known for surfaces by theorem of Bogomolov-Mumford. Initiated by the work of \textit{K. Oguiso} [Int. J. Math. 4, No. 3, 439--465 (1993; Zbl 0793.14030)] and \textit{P. M. H. Wilson} [Invent. Math. 98, No. 1, 139--155 (1989; Zbl 0688.14032)] many authors have discussed the existence of rational curves on higher-dimensional Calabi-Yau manifolds with special structures, for example if \(X\) admits a fibration. In this paper the author starts by studying the existence of rational curves for projective varieties with klt singularities admitting an elliptic fibration \(f: X \rightarrow B\) such that the canonical class is the pull-back of a Cartier divisor from the base. He proves that if \(X\) does not admit a holomorphic one-form, even after finite étale cover, then \(X\) contains a uniruled divisor. In particular the statement applies to singular Calabi-Yau spaces. The author gives an interesting application of his theorem: let \(X\) be a Calabi-Yau space of dimension at least three admitting a nef divisor \(D\) such that \(c_1(D)^2=0\) and \(c_1(D) \cdot c_2(X) = 0\). Then \(X\) contains a rational curve. Note that \(D\) is expected to be semiample but this is a difficult open problem, even for threefolds. For the proof the author shows that \(X\) also admits a divisor \(N\) that is globally generated and defines an elliptic fibration on \(X\).On the constant scalar curvature Kähler metrics. I: A priori estimateshttps://zbmath.org/1472.140422021-11-25T18:46:10.358925Z"Chen, Xiuxiong"https://zbmath.org/authors/?q=ai:chen.xiuxiong"Cheng, Jingrui"https://zbmath.org/authors/?q=ai:cheng.jingruiThis groundbreaking paper is a technical tour de force on the constant scalar curvature Kähler (CSCK) equation. The basic problem is Calabi's dream of finding canonical metric representatives inside positive line bundle classes. A prototype result is Yau's solution to the Calabi conjecture, and the Chen-Donaldson-Sun solution of the Kähler-Einstein equation in the Fano case subject to K-stability. While a formal infinite dimensional GIT framework has long pointed towards CSCK metrics as the natural generality to consider the canonical metric problem, there are significant technical hurdles to go beyond the Kähler-Einstein case, most notably because the CSCK equation is 4th order instead of second order, and because one loses a priori Ricci curvature bounds, which are essential for applying Cheeger-Colding theory. The goal of this paper is to address the central PDE difficulties. Its techniques are relatively classical, involving maximum principles, Moser style iterations, and Alexandrov maximum principles; the presentation is largely self contained, and accessible to those with basic knowledge of Kähler geometry. However, the application of these techniques are often very clever, and exploit a number of rather delicate cancellation effects which can only be appreciated through substantial calculations.
The 4th order CSCK equation on a compact Kähler manifold can be written as a second order coupled system. The authors consider a slightly more general system (needed for their further work on the continuity method)
\[
\log \det(g_{i\bar{j}}+ \phi_{i\bar{j}}) = F + \log \det(g_{i\bar{j}}),\ \Delta_\phi F = -f + Tr_\phi \eta. \tag{1}
\]
When $f = -R$ (the average Ricci scalar) and $\eta = Ric_g$ this is the CSCK equation. The first equation is complex Monge-Ampère, and the second amounts to a prescription of scalar curvature. The main result of this paper is that under an a priori entropy bound $\int e^FFd\mathrm{vol}_g \leq C$, then the solution to this PDE system is bounded to all derivatives. The entropy bound is natural to the problem, because the CSCK equation is the critical point of the Mabuchi functional (also called the K-energy), which can be written as the entropy term plus a well behaved pluripotential term. The stability condition should be thought of as a coercivity condition on the Mabuchi functional, which will essentially force a bound on the entropy as explained in the subsequent works of the authors in the series.
Some highlights of this paper are:
\begin{itemize}
\item In Section 5, the authors prove a $C^0$-estimate on the Kähler potential from the entropy bound. The main ingredient is an ingenious application of the Alexandrov maximum principle, with the additional input of the Skoda inequality. This is inspired by earlier work of Blocki. Even though the method is quite classical, this result is surprisingly strong, especially in the light of Kolodziej's celebrated $L^\infty$-potential estimate from an a priori $L^p$ bound on $F$ with $p > 1$. This part is of significant independent interest in Kähler geometry, especially the analysis of complex Monge-Ampère.
\item In Section 2, they prove among others things an a priori gradient bound $|\nabla \phi |^2e^{-F} \leq C$. This uses a maximum principle argument, which involves a delicate cancellation effect to knock out some bad mixed derivative terms in the Laplacian.
\item In Section 3, they prove a $W^{2,p}$ type estimate $\int e^{-\alpha(p)F}(n+ \Delta \phi )^pd\mathrm{vol}_g \leq C(p)$ for any exponent $p > 0$. This involves integration by part and an iteration argument. The key is that one can gain exponents on $n+ \Delta \phi$ from the nonlinearity, and any derivative terms of $F$ from the Laplacian computation can be either estimated away in complete squares, or absorbed into the equation for $\Delta F$ which is then a priori controlled.
\item In Section 4, they prove simultaneously $|\nabla F| \leq C$ and $n + \Delta \phi \leq C$, whence the metric has $C^2$ bounds, which reduces the CSCK problem to well known higher order estimates. This is proved by a Moser iteration style argument, based on the $W^{2,p}$ estimate established earlier. The reason for $|\nabla F|^2$ to feature in the proof of the metric upper bound $n + \Delta \phi \leq C$, is that one needs the Laplacian of $|\nabla F|^2$ to provide good Hessian terms. The maximum principle quantity involves another subtle cancellation effect to knock out some bad mixed derivative terms. The reason for the Moser iteration to work, is that Sobolev inequality improves the Lebesgue exponent by a definite magnifying factor > 1, while the fact that \(p\) can be arbitrarily large in the $W^{2,p}$ estimate, ultimately ensures that the application of Hölder inequality can only worsen the Lebesgue exponent by a factor which is arbitrarily close to one, so in the end the improvement effect will win over.
\end{itemize}On the anti-canonical geometry of weak \(\mathbb{Q}\)-Fano threefolds. IIhttps://zbmath.org/1472.140432021-11-25T18:46:10.358925Z"Chen, Meng"https://zbmath.org/authors/?q=ai:chen.meng"Jiang, Chen"https://zbmath.org/authors/?q=ai:jiang.chenSummary: By a canonical (resp. terminal) weak \(\mathbb{Q}\)-Fano \(3\)-fold we mean a normal projective one with at worst canonical (resp. terminal) singularities on which the anti-canonical divisor is \(\mathbb{Q}\)-Cartier, nef and big. For a canonical weak \(\mathbb{Q}\)-Fano \(3\)-fold \(V\), we show that there exists a terminal weak \(\mathbb{Q}\)-Fano 3-fold \(X\), being birational to \(V\), such that the \(m\)-th anti-canonical map defined by \(\vert -mK_X\vert\) is birational for all \(m\geq 52\). As an intermediate result, we show that for any \(K\)-Mori fiber space \(Y\) of a canonical weak \(\mathbb{Q}\)-Fano 3-fold, the \(m\)-th anti-canonical map defined by \(\vert -mK_Y\vert\) is birational for all \(m\geq 52\).
For Part I, see [the authors, J. Differ. Geom. 104, No. 1, 59--109 (2016; Zbl 1375.14137)].Fano manifolds and stability of tangent bundleshttps://zbmath.org/1472.140442021-11-25T18:46:10.358925Z"Kanemitsu, Akihiro"https://zbmath.org/authors/?q=ai:kanemitsu.akihiroLet \(X\) be a Fano manifold, that is a complex projective manifold such that the anticanonical divisor \(-K_X\) is ample. If the Picard number of \(X\) is one, a widely believed folklore conjecture claims that the tangent bundle \(T_X\) is semistable (in the sense of Mumford-Takemoto). In this paper the author gives a series of counterexamples to this conjecture! \newline The counterexamples are obtained by a family of horospherical varieties classified by \textit{B. Pasquier} [Math. Ann. 344, No. 4, 963--987 (2009; Zbl 1173.14028)]: for these manifolds the action of the group \(\mbox{Aut}^0(X)\) on \(X\) has two orbits, the open orbit \(X^0\) and a closed orbit \(Z\). Moreover the action on the blow-up \(\mbox{Bl}_Z X\) again has two orbits, the open orbit \(X^0\) and the exceptional divisor \(E\). The manifold \(\mbox{Bl}_Z X\) admits a smooth fibration onto a lower-dimensional manifold \(Y\), the push-forward of the relative tangent bundle to \(X\) defines an algebraically integrable foliation \(\mathcal F \subset T_X\). The author shows that this foliation is canonical in the sense that it is the unique algebraically integrable foliation on \(X\) that is \(\mbox{Aut}^0(X)\)-invariant. General arguments show that the stability of \(T_X\) can be verified by computing the slope of the foliation \(\mathcal F\). It turns out that for infinitely many manifolds in Pasquier's list, the subsheaf \(\mathcal F \subset T_X\) destabilises the tangent bundle. The reviewer recommends to any complex geometer to read this beautiful paper.An effective bound for reflexive sheaves on canonically trivial 3-foldshttps://zbmath.org/1472.140452021-11-25T18:46:10.358925Z"Vermeire, Peter"https://zbmath.org/authors/?q=ai:vermeire.peterLet \(X\) be a smooth projective threefold and \(F\) be a rank 2 reflexive sheaf on \(X\). Starting with \textit{R. Hartshorne}'s classical article [Math. Ann. 254, 121--176 (1980; Zbl 0431.14004)], several bounds for \(c_3(F)\) have been derived in various settings, among them, by the author in the case $\mathrm{Pic}(X)=\mathbb Z L\) for \(L\)-semistable \(F\) [Pac. J. Math. 219, No. 2, 391--398 (2005; Zbl 1107.14032)] in terms of \(c_1(X)\), \(c_2(X)\), \(c_1(F)\) and \(c_2(F)\) .
\textit{A. Gholampour} and \textit{M. Kool} [J. Pure Appl. Algebra 221, No. 8, 1934--1954 (2017; Zbl 06817567)] conjectured that this remains true for any smooth projective threefold. In the paper under review, the author gives explicit effective bounds in the case of a polarized smooth projective threefold \(X\) with \(\omega_X=\mathcal O_X\) (and general Picard group).Primitive elements for \(p\)-divisible groupshttps://zbmath.org/1472.140462021-11-25T18:46:10.358925Z"Kottwitz, Robert"https://zbmath.org/authors/?q=ai:kottwitz.robert-edward"Wake, Preston"https://zbmath.org/authors/?q=ai:wake.prestonLet \(\mathcal G\) be a finite flat group scheme. Using Raynaud's theory of Haar measures of [\textit{M. Raynaud}, Bull. Soc. Math. Fr. 102, 241--280 (1974; Zbl 0325.14020)], the authors define a closed subscheme \(\mathcal G^\times\hookrightarrow\mathcal G\) by decreeing its ideal to be the line bundle of invariant measures on the Cartier dual of \(\mathcal G\). Moreover, if \(\mathcal G\) is a Barsotti-Tate group of level \(n\) for a prime number \(p\), the authors define the closed subscheme \(\mathcal G^{prim}\hookrightarrow\mathcal G\) of ``primitive elements'' to be the inverse image of \(\mathcal G[p]^\times\) under the canonical epimorphism \(\mathcal G\twoheadrightarrow\mathcal G[p]\).
This concept paves the way for a new approach to integral models \(\mathfrak X\) of Shimura varieties with level \(\Gamma_1(p^n)\)-structure, simply by looking at the scheme \(\mathcal A[p^n]^{prim}\) where \(\mathcal A\) is the universal abelian \(g\)-fold (possibly with additional structure). These schemes are not necessarily normal, but they are still interesting counterweights to the more classical concept of moduli spaces of points ``with exact order \(p^n\)'', which seems to be limited to settings in which one can work with one-dimensional formal groups, as in [\textit{V. G. Drinfel'd}, Math. USSR, Sb. 23, 561--592 (1976; Zbl 0321.14014); translation from Mat. Sb., n. Ser. 94(136), 594--627 (1974)] or [\textit{N. M. Katz} and \textit{B. Mazur}, Arithmetic moduli of elliptic curves. Princeton, NJ: Princeton University Press (1985; Zbl 0576.14026)].An analytic application of Geometric Invariant Theoryhttps://zbmath.org/1472.140472021-11-25T18:46:10.358925Z"Buchdahl, Nicholas"https://zbmath.org/authors/?q=ai:buchdahl.nicholas-p"Schumacher, Georg"https://zbmath.org/authors/?q=ai:schumacher.georgOne of the mathematical highlights of the 1980s is the establishment of the Hitchin-Kobayashi correspondence. Given a holomorphic vector bundle over a compact Kähler manifold, this correspondence states that the bundle admits a Hermite-Einstein metric if and only if it satisfies the algebro-geometric condition of slope polystability [\textit{S. K. Donaldson}, Proc. Lond. Math. Soc. (3) 50, 1--26 (1985; Zbl 0529.53018); \textit{K. Uhlenbeck} and \textit{S. T. Yau}, Commun. Pure Appl. Math. 39, S257--S293 (1986; Zbl 0615.58045)].
Almost equally important as the statement is the following expectation that arises from this correspondence. The notion of slope stability was introduced in relation to considerations from moduli spaces, and so one should be able to form moduli spaces of holomorphic vector bundles which (equivalently) admit a Hermite-Einstein metric or are slope polystable. This expectation is well understood when the compact Kähler manifold is actually a smooth projective variety through work of many authors, such as \textit{C. T. Simpson} [Publ. Math., Inst. Hautes Étud. Sci. 79, 47--129 (1994; Zbl 0891.14005)] and \textit{D. Greb} et al. [Geom. Topol. 25, No. 4, 1719--1818 (2021; Zbl 07379436)].
The paper under review is a key step towards understanding the story beyond the projective setting. The authors use their prior work, demonstrating which deformations of polystable bundles remain polystable, to construct a moduli space of slope polystable bundles over a compact Kähler manifold. They also construct a natural Kähler metric on the moduli space. The method they employ is to view their prior work as producing ``charts'' on the moduli space; this strategy had previously been used in other contexts, but the work of the authors contains some new features.
The paper is very clearly and carefully written, and should be viewed as an important contribution to the field.Satellites of spherical subgroupshttps://zbmath.org/1472.140482021-11-25T18:46:10.358925Z"Batyrev, Victor"https://zbmath.org/authors/?q=ai:batyrev.victor-v"Moreau, Anne"https://zbmath.org/authors/?q=ai:moreau.anneLet \(G\) be a complex, connected, reductive algebraic group and let \(H \subset G\) a spherical subgroup (that is, the Borel subgroups of \(G\) act on \(G/H\) with an open orbit). In the present paper the authors define and study the \textit{satellites} of \(H\). The satellites of \(H\) form a family of spherical subgroups of \(G\) defined up to conjugation. They have the same dimension of \(H\) and they encode important information on \(G/H\) and its equivariant embeddings. If \(r\) denotes the rank of \(G/H\), then \(H\) possesses precisely \(2^r\) satellites, which are naturally parametrized by the subsets of the spherical roots of \(G/H\).
The satellites of a spherical subgroup \(H\) have important connections with the embedding theory of \(G/H\), as they impose strong conditions on the stabilizers which can occurr in the toroidal equivariant embeddings of \(G/H\). The satellites of \(H\) can be defined in terms of the \(G\)-invariant valuations of \(G/H\): every such a valuation \(v\) defines a satellite \(H_v\), and two satellites \(H_v\) and \(H_{v'}\) are conjugated if and only if the two invariant valuations \(v\) and \(v'\) belong to the same open face of the valuation cone of \(G/H\). The subgroups of the shape \(H_v\) are called \textit{Brion subgroups}, as they were first studied by \textit{M. Brion} [J. Algebra 134, No. 1, 115--143 (1990; Zbl 0729.14038)]. The Brion subgroup \(H_v\) can also be described in terms of limits of stabilizers of points in the arc space.
Building on previous work of \textit{M. Brion} and \textit{E. Peyre} [Compos. Math. 134, No. 3, 319--335 (2002; Zbl 1031.13007)], in the last section of the paper the authors study the virtual Poincaré polynomial of the homogeneous varieties \(G/H'\), where \(H'\) is a satellite of \(H\). In particular they state a conjecture on the ratio between the virtual Poincaré polynomial of \(G/H'\) and that of \(G/H\), which is proved if either \(H\) is connected or the rank of \(G/H\) is one.On Fano complete intersections in rational homogeneous varietieshttps://zbmath.org/1472.140492021-11-25T18:46:10.358925Z"Bai, Chenyu"https://zbmath.org/authors/?q=ai:bai.chenyu"Fu, Baohua"https://zbmath.org/authors/?q=ai:fu.baohua"Manivel, Laurent"https://zbmath.org/authors/?q=ai:manivel.laurentComplete intersections in rational homogeneous varieties provide many interesting examples of Fano varieties. It is expected by Hartshorne that all smooth subvarieties \(\mathbb{P}^n\) of small codimension are complete intersections. In this paper, the authors studies two geometrical properties of Fano complete intersections in rational homogeneous varieties: local rigidity and quasi-homogeneity.
\textit{R. Bott} [Ann. Math. (2) 66, 203--248 (1957; Zbl 0094.35701)] shows that \(H^i(G/P, T_{G/P}) = 0\) for all \(i \geq 1\) and any rational homogeneous variety \(G/P\), hence they \(G/P\) is locally rigid by Kodaira-Spencer deformation theory. In [\textit{F. Bien} and \textit{M. Brion}, Compos. Math. 104, No. 1, 1--26 (1996; Zbl 0910.14004)], the local rigidity is proven for Fano regular \(G\)-varieties. The case of two-orbits varieties of Picard number one is studied in [\textit{B. Pasquier} and \textit{N. Perrin}, Math. Z. 265, No. 3, 589--600 (2010; Zbl 1200.14097)].
Let \(G/P\) be a rational homogeneous variety with \(G\) simple and \(X =\cap D_i \subset G/P\) a smooth irreducible complete intersection of \(r\) ample divisors. Suppose that \(K^*_{G/P}\otimes \mathcal{O}_{G/P}(-\sum D_i)\) is ample, which implies that \(X\) is Fano. When \(G/P\) is of Picard number one, the converse holds, but in general this condition is stronger than the Fanoness of X. The main theorem of this paper classifies such \(X\) which are locally rigid. It uses the fact that \(H^i(X, T_X) = 0\) for all \(i \geq 2\) by Kodaira-Nakano vanishing theorem; so \(X\) is locally rigid if and only if \(H^1(X, T_X) = 0\).
The authors explain their theorem using Vinberg's theory of parabolic prehomogeneous spaces [\textit{L. Manivel}, Rend. Semin. Mat., Univ. Politec. Torino 71, No. 1, 35--118 (2013; Zbl 1362.11099)]. Given a connected Dynkin diagram \(D\), with a special node satisfying some conditions, we obtain a simply connected simple Lie group \(G\) and a maximal parabolic subgroup \(P\) with the following property. There is an embedding of \(G/P\) in the projectivization \(\mathbb{P}V_P^*\) of a (dualized) fundamental representation, such that \(G \times GL_k\) acts on \(V_P \otimes \mathbb{C}^k\) with finitely many orbits. In particular \(G\) acts on \(Gr(k, V_P )\) with only finitely many orbits, and therefore there exists only a finite number of isomorphism types of codimension \(k\) linear sections of \(G/P\). In this situation, the local rigidity of the general section can be expected, and this is exactly what happens.
It turns out that most of the varieties obtained in the main theorem are hyperplane sections. The authors classify general hyperplane sections which are quasi-homogeneous. An observation is that a general hyperplane section of \(G/P\) is quasi-homogeneous if and only if it is locally rigid but not a hyperplane section of \(Gr(3, 8)\). In general, there is no direct relation between the two properties.Critical loci in computer vision and matrices dropping rank in codimension onehttps://zbmath.org/1472.140502021-11-25T18:46:10.358925Z"Bertolini, Marina"https://zbmath.org/authors/?q=ai:bertolini.marina"Besana, Gian Mario"https://zbmath.org/authors/?q=ai:besana.gian-mario"Notari, Roberto"https://zbmath.org/authors/?q=ai:notari.roberto"Turrini, Cristina"https://zbmath.org/authors/?q=ai:turrini.cristinaThe main goal of this work is to conclude the analysis of critical loci started in [\textit{M. Bertolini} et al., J. Symb. Comput. 91, 74--97 (2019; Zbl 1403.14070)]. In [loc. cit.] it is shown that the minimal generators of the ideal of the critical locus for 3 projections from \(\mathbb{P}^4\) to \(\mathbb{P}^2\) are cubic polynomials assuming that such minors do not share any common factors. However, in this work the author study the case of 3 projections from \(\mathbb{P}^4\) to \(\mathbb{P}^2\), while dropping the genericity assumptions.
First, the classification of canonical forms of \((n+1)\times n\) matrices, for \(n\leq 3\), of linear forms that drop rank in codimension 1 is introduced. Dropping rank in codimension 1 means that the maximal minors have a non trivial common factor of degree either 1 or 2. Thus, Theorem 2.1 provides all canonical forms of the \(4\times 3\) matrices of linear forms whose maximal minors have a greatest common divisor of degree 1, and Theorem 2.2 when maximal minors have a greatest common divisor of degree 2. Once the classification is done, the authors study the loci where these canonical forms drop rank under some mild generality assumptions and in the main dimensional context of interest for computer vision goals. Thus we arrive at the main theorem of the article, where it is classified the critical locus in the case of three projections from \(\mathbb{P}^4\) to \(\mathbb{P}^2\) in the degenerate case. Finally, last sections introduce the application of their results to the problem of reconstruction in computer vision.Codes on linear sections of the Grassmannianhttps://zbmath.org/1472.140512021-11-25T18:46:10.358925Z"Carrillo-Pacheco, Jesús"https://zbmath.org/authors/?q=ai:carrillo-pacheco.jesus"Zaldívar, Felipe"https://zbmath.org/authors/?q=ai:zaldivar.felipeLet \(\mathbb{F}_q\) denote a finite field having \(q\) elements, where \(q\) is a prime power. For a vector space \(E\) of finite dimension \(m\) over \(\mathbb{F}_q\) and \(\ell\le m\), let \(G(\ell, m)\) denote the Grassmannian variety of vector subspaces of dimension \(\ell\) of \(E\). A projective variety \(X\) is called a linear section of the of the Grassmannian \(G(\ell,m)\) if \(X=G(\ell, m)\cap Z(g_1,\ldots, g_N)\), where \(g_1, g_2, \ldots, g_N\) are linearly independent functionals in the ideal that they generate and \(X(\mathbb{F}_q)=\{P_1, \ldots, P_M\}\) is a non-empty set of \(\mathbb{F}_q\)-rational points of \(X\). In this paper, authors study parity-check codes by showing that for every linear section of a Grassmannian, there exists a parity check code with good properties depending on the linear sections. For the Lagrangian-Grassmannian variety, they reveal that these parity-check codes are the low density parity check (LDPC) codes. They also obtained some properties of parity check codes associated to linear sections of Grassmannians.Maximal indexes of flag varieties for spin groupshttps://zbmath.org/1472.140522021-11-25T18:46:10.358925Z"Devyatov, Rostislav A."https://zbmath.org/authors/?q=ai:devyatov.rostislav-a"Karpenko, Nikita A."https://zbmath.org/authors/?q=ai:karpenko.nikita-a"Merkurjev, Alexander S."https://zbmath.org/authors/?q=ai:merkurjev.alexander-sSummary: We establish the sharp upper bounds on the indexes for most of the twisted flag varieties under the spin groups \({\operatorname{\text{Spin}}(n)} \).Cotangent bundles of partial flag varieties and conormal varieties of their Schubert divisorshttps://zbmath.org/1472.140532021-11-25T18:46:10.358925Z"Lakshmibai, V."https://zbmath.org/authors/?q=ai:lakshmibai.venkatramani"Singh, R."https://zbmath.org/authors/?q=ai:singh.reetendra|singh.rupam|singh.robin-vikram|singh.rishi-pal|singh.ranjeet-k|singh.ranveer-kumar|singh.rajrupa|singh.ravinder.2|singh.ram-gopal|singh.reshma|singh.ram-sakal|singh.rajkeshar|singh.r-t|singh.raghvendra|singh.rishi-ranjan|singh.rakhi|singh.rajiv-r-p|singh.raj-narayan|singh.rekha|singh.rupen-pratap|singh.rajnish-kumar|singh.radhey-s|singh.r-k-tarachand|singh.ranjit-j|singh.ravi-p|singh.raman-p|singh.rajpal|singh.ram-chandra|singh.renu|singh.ratna|singh.rajeshwar-prasad|singh.rishi-n|singh.rajwinder|singh.rita|singh.rajiv-kumar|singh.rajmeet|singh.ravendra|singh.rajesh-kumar|singh.ranveer|singh.roli|singh.rajbir|singh.ran-vir|singh.rajendra-prasad|singh.rajkumar-brojen|singh.r-k-p|singh.ram-veer|singh.ramji|singh.raushan|singh.rajindar|singh.ram-narayan|singh.rajani|singh.raghuraj|singh.rajinder-pal|singh.r-d|singh.ramendra-p|singh.rajat-kumar|singh.ritambhara|singh.raj-narain|singh.ram-nawal|singh.rasphal|singh.ram-kishore|singh.rattandeep|singh.rohit-r|singh.rajesh-pratap|singh.ruchi|singh.rattan|singh.ramkaran|singh.robby|singh.raghavendra|singh.rameshwar|singh.reen-nripjeet|singh.rana-p|singh.rajeev-kumar|singh.rahul-kumar|singh.raghuvansh-p|singh.ram-mehar|singh.randhir|singh.rama-krishna|singh.ram-binoy|singh.raghu-nath|singh.rishabh|singh.rajvir|singh.ramanpreet|singh.rajdeep|singh.rituraj|singh.raj-kishor|singh.rayman-preet|singh.ramesh-kumar|singh.ram-naresh|singh.r-a|singh.ramanand|singh.ronal|singh.richa|singh.rakeshwar|singh.rashmi|singh.rangi|singh.randheer|singh.rishi-ram-h-n|singh.ravindra-pratap|singh.ramandeep|singh.rajneesh-kumar|singh.ram-nandan-p|singh.roshani|singh.ripudaman|singh.ranbirSummary: Let \(P\) be a parabolic subgroup in \(G = \mathrm{SL}_n(\mathbf k)\), for \textbf{k} an algebraically closed field. We show that there is a \(G\)-stable closed subvariety of an affine Schubert variety in an affine partial flag variety which is a natural compactification of the cotangent bundle \(T^* G/P\). Restricting this identification to the conormal variety \(N^* X(w)\) of a Schubert divisor \(X(w)\) in \(G/P\), we show that there is a compactification of \(N^* X(w)\) as an affine Schubert variety. It follows that \(N^* X(w)\) is normal, Cohen-Macaulay, and Frobenius split.Fujita's freeness conjecture for \(T\)-varieties of complexity onehttps://zbmath.org/1472.140542021-11-25T18:46:10.358925Z"Altmann, Klaus"https://zbmath.org/authors/?q=ai:altmann.klaus"Ilten, Nathan"https://zbmath.org/authors/?q=ai:ilten.nathan-owenFor a not too singular projective variety \(X\) and an ample divisor \(H\) on it, \textit{T. Fujita} conjectured in [Adv. Stud. Pure Math. 10, 167--178 (1987; Zbl 0659.14002)] that \(mH+K_X\) is basepoint free, if \(m > \dim(X)\).
This conjecture holds for curves by the Riemann-Roch theorem. If one asks only for nefness, then the conjecture holds for any \(X\) with at most rational Gorenstein singularities, as shown by Fujita in [loc. cit.]. As a nef divisor on a toric variety is automatically basepoint free, the original conjecture holds for Gorenstein toric varieties.
Based on these two results, it is quite natural to ask, whether the conjecture holds for \(T\)-varieties of complexity one with at most rational Gorenstein singularities. Such a variety \(X\) comes with an effectiv action of a torus \(T\) with \(\dim(T) = \dim(X)-1\), so the Chow quotient \(Y = X/T\) is a curve and a generic fiber a toric variety, see [\textit{K. Altmann} et al., in: Contributions to algebraic geometry. Impanga lecture notes. Based on the Impanga conference on algebraic geometry, Banach Center, Bȩdlewo, Poland, July 4--10, 2010. Zürich: European Mathematical Society (EMS). 17--69 (2012; Zbl 1316.14001)].
The central result of this article is that in this case, Fujita's Freeness Conjecture holds.
For the proof, note that for general \(T\)-varieties, nefness does not automatically imply basepoint freeness. Indeed the authors show that implication only for nef divisors of the form \(mH+K_X\), with \(H\) ample and \(m > \dim(X)\). Moreover, for \(k \in \mathbb N\), they give a sequence of smooth \(\mathbb K^*\)-surfaces \(X_k\) (the simplest \(T\)-varieties of complexity one) with ample divisor \(H_k\) such that \(kH_k\) is not basepoint free (but are nef as soon as \(k>2\)).Toric co-Higgs sheaveshttps://zbmath.org/1472.140552021-11-25T18:46:10.358925Z"Altmann, Klaus"https://zbmath.org/authors/?q=ai:altmann.klaus"Witt, Frederik"https://zbmath.org/authors/?q=ai:witt.frederikFor a complex toric variety \(X_{\Sigma}\) given by a fan \(\Sigma\subseteq N_{\mathbb R}=N\otimes_{\mathbb Z}{\mathbb R}\) for a lattice \(N\) and acting torus \(T=N\otimes_{\mathbb Z}{\mathbb C}\), by \textit{A. Klyachko} [Math. USSR, Izv. 35, No. 2, 337--375 (1990; Zbl 0706.14010)] a \textit{toric sheaf} \({\mathcal E}\), (that is, an \({\mathcal O}_X\)-module with an action of the torus \(T\) which is linear on the fibers and is compatible with the \(T\)-action of \(X\)) corresponds to the complex vector space \(E={\mathcal E}_1/{\mathfrak m}_{X,1}{\mathcal E}_1\) (where \({\mathcal E}_1\) is the stalk or \({\mathcal E}\) at \(1\in T\subseteq X\) and \({\mathfrak m}_{X,1}\) is the maximal ideal of \({\mathcal O}_{X,1}\)) together with a decreasing \({\mathbb Z}\)-filtration \(E^{\bullet}_{\rho}\) indexed by the rays \(\rho\in \Sigma(1)\). Following \textit{I. Biswas} et al., [Ill. J. Math. 65, No. 1, 181--190 (2021; Zbl 1465.14048)] a \textit{toric co-Higgs sheaf} is a pair \(({\mathcal E},\Phi)\) consisting of a toric sheaf \({\mathcal E}\) on a toric variety \(X_{\Sigma}\) and a \textit{Higgs field}, i.e., a \(T\)-equivariant \({\mathcal O}_X\)-morphism \(\Phi:{\mathcal E}\rightarrow{\mathcal E}\otimes_{{\mathcal O}_X}{\mathcal T}_X\) such that \(\Phi\wedge\Phi=0\), where \({\mathcal T}_X\) is the tangent sheaf of \(X\). Dropping the integrability condition \(\Phi\wedge\Phi=0\), the pair \(({\mathcal E},\Phi)\) is called a \textit{toric pre-co-Higgs sheaf} and \(\Phi\) a \textit{pre-co-Higgs field}.
The paper under review considers general co-Higgs fields and not only \(M\)-homogeneous ones as in [Biswas et al., loc. cit]. To study these general co-Higgs sheaves, the authors start characterizing pre-co-Higgs fields using Klyachko's formalism by means of associated contractions in Theorem 8 and then show that every co-Higgs field defines a commutative finitely generated \({\mathbb C}[M]\)-algebra, the \textit{Higgs algebra}. Next, the authors introduce some combinatorial invariants: First, using that a pre-co-Higgs field is a direct sum \(\Phi=\sum \Phi^r\) of maps \(\Phi^r:{\mathcal E}\rightarrow {\mathcal E}\otimes_{{\mathcal O}_X}{\mathcal T}_X\) of degree \(r\in M\), they define the corresponding \textit{Higgs polytope} of \(\Phi\) as the convex hull in \(M_{\mathbb R}\) of its support \(\text{supp}(\Phi)=\{r\in M: \Phi^r\neq 0\}\subseteq M\). The convex hull of the totality of degrees \(r\in M\) of all possible toric pre-co-Higgs fields defines a second polytope, the \textit{Higgs range}. After proving some properties of these polytopes, in the last two sections of the paper they are calculated for several smooth toric surfaces: They compute the Higgs range of the projective plane and Hirzebruch and Fano surfaces, and they also compute the Higgs polytope for some del Pezzo surfaces. The whole paper includes many illustrative examples with explicit calculations nicely complementing the developments.Strong factorization and the braid arrangement fanhttps://zbmath.org/1472.140562021-11-25T18:46:10.358925Z"Machacek, John"https://zbmath.org/authors/?q=ai:machacek.john-mSummary: We establish strong factorization for pairs of smooth fans which are refined by the braid arrangement fan. Our method uses a correspondence between cones and preposets.Singular curves of low degree and multifiltrations from osculating spaceshttps://zbmath.org/1472.140572021-11-25T18:46:10.358925Z"Buczyński, Jarosław"https://zbmath.org/authors/?q=ai:buczynski.jaroslaw"Ilten, Nathan"https://zbmath.org/authors/?q=ai:ilten.nathan-owen"Ventura, Emanuele"https://zbmath.org/authors/?q=ai:ventura.emanueleSummary: In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree \(d\) rational curves in \(\mathbb{P}^n\) when \(d-n\leq 3\) and \(d<2n\). Along the way, we describe the Schubert cycles giving rise to these projections. We also reprove a special case of the Castelnuovo bound using these multifiltrations: under the assumption \(d<2n\), the arithmetic genus of any non-degenerate degree \(d\) curve in \(\mathbb{P}^n\) is at most \(d-n\).Syzygies of the apolar ideals of the determinant and permanenthttps://zbmath.org/1472.140582021-11-25T18:46:10.358925Z"Alper, Jarod"https://zbmath.org/authors/?q=ai:alper.jarod"Rowlands, Rowan"https://zbmath.org/authors/?q=ai:rowlands.rowanGiven a polynomial \(f\in \mathbb K[y_1,\ldots,y_k]\) one defines its apolar ideal \(f^{\bot}\) as \[f^{\bot}=\{g\in\mathbb K[y_1,\ldots,y_k] : \partial g(f)=0\}.\] Recall that to a monomial \(y^{\alpha}=y_1^{\alpha_1}\ldots y_k^{\alpha_k}\) one associates a differential operator \[\frac{\partial}{\partial y^{\alpha}}=\frac{\partial}{\partial y_1^{\alpha_1}\cdots\partial y_k^{\alpha_k}}\] and extends this definition linearly to all polynomials.
Thus for \(g=\sum c_{\alpha}y^{\alpha}\) one associates a differential operator \[\partial g= \sum c_{\alpha}\frac{\partial}{\partial y^{\alpha}}.\] The authors of the paper under review are interested in apolar ideals of two specific polynomials \(\mathrm{def}_n\) and \(\mathrm{perm}_n\) which are elements of the ring \(\mathbb K[x_{11},\ldots, x_{1n},x_{21},\ldots,x_{nn}]\) defines as \[\mathrm{def}_n=\sum_{\sigma\in S_n} \mathrm{sgn}(\sigma) x_{1\sigma(1)}\ldots x_{n\sigma(n)}\] and \[\mathrm{perm}_n=\sum_{\sigma\in S_n} x_{1\sigma(1)}\ldots x_{n\sigma(n)}.\] [\textit{S. M. Shafiei}, J. Commut. Algebra 7, No. 1, 89--123 (2015; Zbl 1364.13024)] showed that the ideals \((\mathrm{def}_n)^{\bot}\) and \((\mathrm{perm}_n)^{\bot}\) are generated by quadrics. She provided an explicit minimal set of generators. The authors extend this study to the first syzygies. They show that the first syzygies of \((\mathrm{def}_n)^{\bot}\) are linear except in characteristic two, where both polynomials and hence their apolar ideals coincide. Thus \((\mathrm{def}_n)^{\bot}\) satisfies at lest the \(N_3\) property of \textit{M. L. Green} [J. Differ. Geom. 19, 125--167, 168--171 (1984; Zbl 0559.14008)].
On the other hand syzygies of \((\mathrm{perm}_n)^{\bot}\) require also quadratic generators, in arbitrary characteristic. Thus one can distinguish both polynomials by properties of their minimal graded free resolution.
The paper is clearly written and all arguments are kept pretty effective, even if some of them are quite involved.On the Terracini locus of projective varietieshttps://zbmath.org/1472.140592021-11-25T18:46:10.358925Z"Ballico, Edoardo"https://zbmath.org/authors/?q=ai:ballico.edoardo"Chiantini, Luca"https://zbmath.org/authors/?q=ai:chiantini.lucaThe aim of this article is to introduce and study the Terracini locus of \((X,L)\), where \(X\) is a projective variety and \(L\) is a line bundle on \(X\).
Let \(X \subset \mathbb{P}^N\) be a smooth integral projective variety of dimension \(n\), and let \(L\) be a line bundle on \(X\). If \(S \subset X_{reg}\) is reduced finite set, then denote with \(2S\) the union of fat points \(2p\) for any \(p \in S\). The \(r\)-Terracini locus of \((X,L)\) is the locally closed set given by all the \(S \subset X_{reg}\) such that \(S\) is reduced and of cardinality \(r\) and \(h^{0}(\mathcal{I}_{(2S,X)}\otimes L) > h^0(L)-(n+1)r\). After recalling some basic facts about \(0\)-dimensional schemes, in the third section the authors provide sufficient conditions to get that a reduced ad finite set \(S \subset X_{reg}\) is not in the \(r\)-th Terracini locus, also in the case of \(L = L_1 \otimes L_2\), with \(L_i\) line bundles. The fourth and fifth sections are devoted to show examples when \(X\) is a projective space and \(L = \mathcal{O}(d)\). Moreover, bounds on the dimension of the Terracini locus are provided. Eventually the authors underline the link between the study of the Terracini locus and the \(r\)-identifiability of a point \(u\) of a secant variety \(S_r(X)\), i.e. the uniqueness of the decomposition of \(u\) in sum of \(r\) points of \(X\).Conjecture \(\mathcal{O}\) holds for some horospherical varieties of Picard rank 1https://zbmath.org/1472.140602021-11-25T18:46:10.358925Z"Bones, Lela"https://zbmath.org/authors/?q=ai:bones.lela"Fowler, Garrett"https://zbmath.org/authors/?q=ai:fowler.garrett"Schneider, Lisa"https://zbmath.org/authors/?q=ai:schneider.lisa"Shifler, Ryan M."https://zbmath.org/authors/?q=ai:shifler.ryan-mSummary: Property \(\mathcal{O}\) for an arbitrary complex, Fano manifold \(X\) is a statement about the eigenvalues of the linear operator obtained from the quantum multiplication of the anticanonical class of \(X\). Conjecture \(\mathcal{O}\) is a conjecture that property \(\mathcal{O}\) holds for any Fano variety. Pasquier classified the smooth nonhomogeneous horospherical varieties of Picard rank 1 into five classes. Conjecture \(\mathcal{O}\) has already been shown to hold for the odd symplectic Grassmannians, which is one of these classes. We will show that conjecture \(\mathcal{O}\) holds for two more classes and an example in a third class of Pasquier's list. Perron-Frobenius theory reduces our proofs to be graph-theoretic in nature.Gromov-Witten theory with derived algebraic geometryhttps://zbmath.org/1472.140612021-11-25T18:46:10.358925Z"Mann, Etienne"https://zbmath.org/authors/?q=ai:mann.etienne"Robalo, Marco"https://zbmath.org/authors/?q=ai:robalo.marcoSummary: In this survey we add two new results that are not in our paper [Geom. Topol. 22, No. 3, 1759--1836 (2018; Zbl 1423.14320)]. Using the idea of brane actions discovered by Toën, we construct a lax associative action of the operad of stable curves of genus zero on a smooth variety \(X\) seen as an object in correspondences in derived stacks. This action encodes the Gromov-Witten theory of \(X\) in purely geometrical terms.
For the entire collection see [Zbl 1471.14005].Moduli of stable maps in genus one and logarithmic geometry. Ihttps://zbmath.org/1472.140622021-11-25T18:46:10.358925Z"Ranganathan, Dhruv"https://zbmath.org/authors/?q=ai:ranganathan.dhruv"Santos-Parker, Keli"https://zbmath.org/authors/?q=ai:santos-parker.keli"Wise, Jonathan"https://zbmath.org/authors/?q=ai:wise.jonathanThe paper under review studies moduli space of genus one curves in terms of logarithmic structure. The authors present a smooth and proper construction for compactification of moduli space of stable maps from genus one curves to Kontsevich space. Based on the key Vakil-Zinger's construction of desingularization of the principal component, the authors give a modular understanding their construction by minor modification and develop an analogue construction of moduli space of genus 1 pointed stable quasimaps to Kontsevich space. Especially, the authors develop new techniques to consider the geometry of elliptic \(m\)-fold singularity which results in a modular factorization of the birational maps connecting Smyth's space of genus one curve with marked points. Since there is no modular explanation for blowups of moduli space, so the authors also show how to build a modular explanation for logarithmic blowups of logarithmic moduli spaces by adding tropical information to moduli problem. In summary, the authors apply the techniques of tropical geometry and logarithmic structure to study moduli spaces of stable maps (or quasimaps) from genus one curves to Kontsevich space. It is an interesting exploration about the relationship between tropical geometry, logarithmic moduli space, stable maps and moduli space of elliptic curves.Symmetric non-negative forms and sums of squareshttps://zbmath.org/1472.140632021-11-25T18:46:10.358925Z"Blekherman, Grigoriy"https://zbmath.org/authors/?q=ai:blekherman.grigoriy"Riener, Cordian"https://zbmath.org/authors/?q=ai:riener.cordianThe authors study \textit{symmetric} nonnegative homogeneous polynomials and relationships between the cone of symmetric sums of squares and the cone of symmetric nonnegative forms of fixed degree \(2d\) (in arbitrary numbers of variables).
They provided a uniform representation of the cone of symmetric sums of squares and its dual cone in terms of linear matrix polynomials. In particular, by using the representation, they completely characterize the sums of squares cone \(\Sigma_{n,4}\) of degree \(4\) in \(n\) variable and its boundary, and therefore certify the difference between symmetric sums of squares and symmetric non-negative quartics.
Also, they investigated the asymptotic behavior of the cone of sums of squares and nonnegative forms of fixed degree \(2d\) as the number \(n\) of variables grows. In detail, they showed that the difference between the cone of symmetric nonnegative forms and sums of squares does not grow arbitrarily large for any fixed degree \(2d\) (even though the (volume) difference between the two cones increases exponentially for any even degree \(2d\) as the number \(n\) of variables grows). In particular, they show that the cone of symmetric non-negative quartics and the cone of quartic symmetric sums of squares asymptotically become closer as the number of variables grows by proving the two cones approach the same limit (in degree \(4\)). They conjectured that the limits agree in any degree \(2d\) for \(d>2\).Exponential-constructible functions in \(P\)-minimal structureshttps://zbmath.org/1472.140642021-11-25T18:46:10.358925Z"Chambille, Saskia"https://zbmath.org/authors/?q=ai:chambille.saskia"Cubides Kovacsics, Pablo"https://zbmath.org/authors/?q=ai:cubides-kovacsics.pablo"Leenknegt, Eva"https://zbmath.org/authors/?q=ai:leenknegt.evaThere exist many results showing that certain classes of functions are closed under integration; the main result of the present paper is also such a result, for functions on a finite extension \(K\) of the field \(\mathbb Q_p\) of \(p\)-adic numbers. Recall that there is a natural (Haar) measure on such fields \(K\), yielding a notion of integration of functions \(K^n \to \mathbb C\). More precisely, the authors specify certain classes \(\mathcal H\) of such functions which are ``base-stable under integration'', meaning that whenever a function \(f\colon K^{n+1} \to \mathbb C\) lies in \(\mathcal H\), then so does the function \(K^n \to \mathbb C, x \mapsto \int_{K} f(x, y)\,dy\). To be precise, the authors more generally consider functions on certain subsets of \(K^n \times \mathbb Z^m\), where \(\mathbb Z\) is equipped with the counting measure. In this review, I will for simplicity stick to functions on \(K^n\), and I will write \(\mathcal H(K^n)\) for those functions in \(\mathcal H\) which live on \(K^n\).
In many results of that kind (including the one in the present paper), one first fixes a certain first-order language \(\mathcal L\) on \(K\), yielding in particular a notion of \(\mathcal L\)-definable functions from \(K^n \to \mathbb Z\) (where \(\mathbb Z\) is considered as the value group of \(K\)). Then \(\mathcal H(K^n)\) is defined as some algebra generated by certain specific functions which are specified in terms of \(\mathcal L\)-definable functions.
A classical result of that kind is the one by \textit{J. Denef} [Prog. Math. None, 25--47 (1985; Zbl 0597.12021)], where \(\mathcal L\) is the valued field language, and where \(\mathcal H(K^n)\) is the \(\mathbb Q\)-algebra generated by functions of the form \(\alpha\colon K^n \to \mathbb Z\) and \(K^n \to \mathbb Q, x \mapsto q^{\alpha(x)}\), where \(\alpha\) is \(\mathcal L\)-definable. This has then been generalized in two directions: (a) allow other languages \(\mathcal L\); (b) add more generators to \(\mathcal H(K^n)\).
An important generalization of type (b) is to consider classes of functions containing additive characters \(\psi\colon (K, +) \to (\mathbb C^\times, \cdot)\), since those naturally appear in representation theory and in Fourier transforms. In direction (a), it is tempting to believe that one can take \(\mathcal L\) to be any P-minimal language on \(K\); P-minimality is an axiomatic condition implying that \(\mathcal L\)-definable objects are ``tame'' in some geometric sense. Up to recently, assuming P-minimality was not enough to prove the desired closedness under intergration; one additionally had to assume the existence of definable Skolem functions. This assumption has now been dropped. More precisely, in a previous paper, \textit{P. Cubides Kovacsics} and \textit{E. Leenknegt} [J. Symb. Log. 81, No. 3, 1124--1141 (2016; Zbl 1428.03060)] proved closedness under integration for any P-minimal \(\mathcal L\) when \(\mathcal H(K^n)\) is generated by the same kinds of functions as in the above-mentioned result by Denef. In the present paper, they now show that this result also extends to algebras \(\mathcal H(K^n)\) containing additive characters. In particular, the authors had to find the ``right'' algebras \(\mathcal H(K^n)\) (which are not a straight forward generalization of the ones when definable Skolem functions exist).Counting algebraic points in expansions of o-minimal structures by a dense sethttps://zbmath.org/1472.140652021-11-25T18:46:10.358925Z"Eleftheriou, Pantelis E."https://zbmath.org/authors/?q=ai:eleftheriou.pantelis-eThe Pila-Wilkie theorem, which was used in Pila's solution of André-Oort conjecture [\textit{J. Pila}, Ann. Math. 173, No. 3, 1779--1840 (2011; Zbl 1243.14022)], assets that a set definable in an o-minimal structure having many rational points contains an infinite semialgebraic set [\textit{J. Pila} et al., Duke Math. J. 133, No. 3, 591--616 (2006; Zbl 1217.11066)].
This paper extends the Pila-Wilkie theorem to an expansion \(\langle \mathcal R, P \rangle\) of an o-minimal structure \(\mathcal R\) by a dense set \(P\), (1) which is either an elementary substructure of \(\mathcal R\), or (2) it is dcl-independent. Under these assumptions, a set definable in \(\langle \mathcal R, P \rangle\) having many algebraic points contains an infinite set which is \(\emptyset\)-definable in \(\langle \mathcal R, P \rangle\). The definition of algebraic points comes from [\textit{J. Pila}, Selecta Math. N. S. 15, No. 1, 151--170 (2009; Zbl 1218.11068)]. The structures of the form \(\langle \mathcal R, P \rangle\) satisfying the assumptions are introduced in [\textit{A. van den Dries}, Fund. Math. 157, No. 1, 61--78 (1998; Zbl 0906.03036)] and [\textit{A. Dolich} et al., Ann. Pure Appl. Logic 167, No. 8, 684--706 (2016; Zbl 1432.03070)], respectively.
The above theorem is deduced from Pila's result on algebraic points by introducing a generalization of Pila's algebraic part called algebraic trace part. The proof is brief in the case (1). In the case (2), the theorem is reduced to the case in which the set is a cone using the cone decomposition theorem given in [\textit{P. Eleftheriou} et al., Isr. J. Math. 239, No. 1, 435--500 (2020; Zbl 1472.03037)].The set of separable states has no finite semidefinite representation except in dimension \(3\times 2\)https://zbmath.org/1472.140662021-11-25T18:46:10.358925Z"Fawzi, Hamza"https://zbmath.org/authors/?q=ai:fawzi.hamzaGiven integers \(n \ge m,\) let \(\mathrm{Sep}(n, m)\) be the set of {\em separable states} on the Hilbert space \(\mathbb{C}^n \otimes \mathbb{C}^m,\) i.e., \[\mathrm{Sep}(n, m) := \mathbf{conv}\{x x^\dagger \otimes y y^\dagger \ : \ x \in \mathbb{C}^n, |x| = 1, y \in \mathbb{C}^m, |y| = 1\}.\] Here \(x^\dagger\) indicates conjugate transpose, \(|x|^2 := x^\dagger x\) and \(\mathbf{conv}\) denotes the convex hull.
We say that a convex set \(C \subset \mathbb{R}^d\) has a {\em semidefinite representation} (of size \(r\)) if it can be expressed as \(C = \pi(S),\) where \(\pi \colon \mathbb{R}^D \to \mathbb{R}^d\) is a linear map and \(S \subset \mathbb{R}^D\) is a convex set defined using a linear matrix inequality \[S = \{w \in \mathbb{R}^D \ : \ M_0 + w_1M_1 + \cdots + w_DM_D \succeq 0\}\] where \(M_0, \ldots, M_D\) are Hermitian matrices of size \(r \times r.\)
It is known, from the earlier work of [\textit{S. L. Woronowicz}, Rep. Math. Phys. 10, 165--183 (1976; Zbl 0347.46063)] that for \(n + m \le 5,\) the set \(\mathrm{Sep}(n, m)\) is just the set of states which have a positive partial transpose, and hence it has a semidefinite representation.
In the paper under review, the author shows that for \(n + m > 5,\) the set \(\mathrm{Sep}(n, m)\) has no semidefinite representation, and so this provides a new counterexample to the Helton-Nie conjecture [\textit{J. W. Helton} and \textit{J. Nie}, SIAM J. Optim. 20, No. 2, 759--791 (2009; Zbl 1190.14058)], which was recently disproved by \textit{C. Scheiderer} [SIAM J. Appl. Algebra Geom. 2, No. 1, 26--44 (2018; Zbl 1391.90462)].
The paper is very clear, well written and quite interesting.Differentiable approximation of continuous semialgebraic mapshttps://zbmath.org/1472.140672021-11-25T18:46:10.358925Z"Fernando, José F."https://zbmath.org/authors/?q=ai:fernando.jose-f"Ghiloni, Riccardo"https://zbmath.org/authors/?q=ai:ghiloni.riccardoThis paper concerns the problem of approximating a uniformly continuous semialgebraic map \(f: S \to T\) from a compact semialgebraic set \(S\) to an arbitrary semialgebraic set \(T\) by a semialgebraic map \(g: S \to T\) that is differentiable of class \(C^\nu\), where \(\nu\) is a positive integer or \(\infty\). It is known that if \(T\) is an \(C^\nu\) semialgebraic manifold, then arbitrarily good (in the \(C^\nu\)-norm) \(C^\nu\) semialgebraic approximations exist. The authors show that for \emph{any semialgebraic \(T\)}, arbitrarily good \(\nu = 1\) approximations are possible. For \(\nu \geq 2\), they obtain density results when: (1) \(T\) is compact and locally \(C^\nu\) semialgebraically equivalent to a polyhedron, or (2) \(T\) is an open semialgebraic subset of a Nash set. The paper includes a useful review of approximation results in semialgebraic geometry, including discussion of key references.The real polynomial eigenvalue problem is well conditioned on the averagehttps://zbmath.org/1472.140682021-11-25T18:46:10.358925Z"Beltrán, Carlos"https://zbmath.org/authors/?q=ai:beltran.carlos"Kozhasov, Khazhgali"https://zbmath.org/authors/?q=ai:kozhasov.khazhgaliThe paper deals with polynomial eigenvalue problem, namely its condition number is studied.
First, the solution variety \(\mathcal{S}\) is introduced, which turns out to be real algebraic or semialgebraic subset of \(\mathbb{R}^m\setminus\{0\}\times S^1\), the product of the variety of inputs and the variety of outputs endowed with Finsler structures.
The condition number \(\mu(a)\) of a given input \(a\) is the sum of the local condition numbers \(\mu(a,x)\) for all solutions for the input.
If the solution variety \(\mathcal{S}\) is co-called nondegenerate, the formula for the squared condition number is proven. Afterwards the formula is used for the case of polynomial eigenvalue problem and so a new proof of the latter is obtained.Computing the equisingularity type of a pseudo-irreducible polynomialhttps://zbmath.org/1472.140692021-11-25T18:46:10.358925Z"Poteaux, Adrien"https://zbmath.org/authors/?q=ai:poteaux.adrien"Weimann, Martin"https://zbmath.org/authors/?q=ai:weimann.martinIn the paper under review, the authors characterize a class of germs of plane curve singularities, containing irreducible ones, whose equisingularity type can be computed in an expected quasi-linear time with respect to the discriminant valuation of a Weierstrass equation.Prism graphs in tropical plane curveshttps://zbmath.org/1472.140702021-11-25T18:46:10.358925Z"Jacoby, Liza"https://zbmath.org/authors/?q=ai:jacoby.liza"Morrison, Ralph"https://zbmath.org/authors/?q=ai:morrison.ralph"Weber, Ben"https://zbmath.org/authors/?q=ai:weber.benSummary: Any smooth tropical plane curve contains a distinguished trivalent graph called its skeleton. In [Discrete Math. 344, No. 1, Article ID 112161, 19 p. (2021; Zbl 1455.52013)], the second author and \textit{A. K. Tewari} proved that the so-called big-face graphs cannot be the skeleta of tropical curves for genus \(12\) and greater. In this paper we answer an open question they posed to extend their result to the prism graphs, proving that a prism graph is the skeleton of a smooth tropical plane curve precisely when the genus is at most \(11\). Our main tool is a classification of lattice polygons with two points that can simultaneously view all others, without having any one point that can observe all others.Generalizing classical Clifford algebras, graded Clifford algebras and their associated geometryhttps://zbmath.org/1472.150322021-11-25T18:46:10.358925Z"Vancliff, Michaela"https://zbmath.org/authors/?q=ai:vancliff.michaelaSummary: This article is based on a talk given by the author at the \textit{12th International Conference on Clifford Algebras and their Applications in Mathematical Physics}. A generalization, introduced by \textit{T. Cassidy} and the author [J. Lond. Math. Soc., II. Ser. 90, No. 2, 631--636 (2014; Zbl 1303.16029)]
of a classical Clifford algebra is discussed together with connections between that generalization and a generalization of a graded Clifford algebra. A geometric approach to studying the algebras, viewed through the lens of Artin, Tate and Van den Bergh's noncommutative algebraic geometry, is also presented.Hopf algebras and tensor categories. International workshop, Nanjing University, Nanjing, China, September 9--13, 2019https://zbmath.org/1472.160012021-11-25T18:46:10.358925Z"Andruskiewitsch, Nicolás"https://zbmath.org/authors/?q=ai:andruskiewitsch.nicolas"Liu, Gongxiang"https://zbmath.org/authors/?q=ai:liu.gongxiang"Montgomery, Susan"https://zbmath.org/authors/?q=ai:montgomery.susan"Zhang, Yinhuo"https://zbmath.org/authors/?q=ai:zhang.yinhuoPublisher's description: Articles in this volume are based on talks given at the International Workshop on Hopf Algebras and Tensor Categories, held from September 9--13, 2019, at Nanjing University, Nanjing, China.
The articles highlight the latest advances and further research directions in a variety of subjects related to tensor categories and Hopf algebras.
Primary topics discussed in the text include the classification of Hopf algebras, structures and actions of Hopf algebras, algebraic supergroups, representations of quantum groups, quasi-quantum groups, algebras in tensor categories, and the construction method of fusion categories.
The articles of this volume will be reviewed individually.On some algebras associated to genus one curveshttps://zbmath.org/1472.160172021-11-25T18:46:10.358925Z"Fisher, Tom"https://zbmath.org/authors/?q=ai:fisher.tom-aSummary: In [\textit{D. Haile} and \textit{I. Han}, J. Algebra 313, No. 2, 811--823 (2007; Zbl 1121.14022); \textit{J.-M. Kuo}, J. Algebra 330, No. 1, 86--102 (2011; Zbl 1229.16013)], the respective cited authors have studied certain non-commutative algebras associated to a binary quartic or ternary cubic form. We extend their construction to pairs of quadratic forms in four variables, and conjecture a further generalisation to genus one curves of arbitrary degree. These constructions give an explicit realisation of an isomorphism relating the Weil-Châtelet and Brauer groups of an elliptic curve.Embedding of the derived Brauer group into the secondary \(K\)-theory ringhttps://zbmath.org/1472.160202021-11-25T18:46:10.358925Z"Tabuada, Gonçalo"https://zbmath.org/authors/?q=ai:tabuada.goncaloSummary: In this note, making use of the recent theory of noncommutative motives, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injectivity properties: in the case of a regular integral quasi-compact quasi-separated scheme, it is injective; in the case of an integral normal Noetherian scheme with a single isolated singularity, it distinguishes any two derived Brauer classes whose difference is of infinite order. As an application, we show that the aforementioned canonical map is injective in the case of affine cones over smooth projective plane curves of degree \(\geq 4\) as well as in the case of Mumford's (famous) singular surface.Calabi-Yau algebras and the shifted noncommutative symplectic structurehttps://zbmath.org/1472.160242021-11-25T18:46:10.358925Z"Chen, Xiaojun"https://zbmath.org/authors/?q=ai:chen.xiaojun"Eshmatov, Farkhod"https://zbmath.org/authors/?q=ai:eshmatov.farkhodSummary: In this paper we show that for a Koszul Calabi-Yau algebra, there is a shifted bi-symplectic structure in the sense of Crawley-Boevey-Etingof-Ginzburg [\textit{W. Crawley-Boevey} et al., Adv. Math. 209, No. 1, 274--336 (2007; Zbl 1111.53066)], on the cobar construction of its co-unitalized Koszul dual coalgebra, and hence its DG representation schemes, in the sense of Berest-Khachatryan-Ramadoss
[\textit{Y. Berest} et al., Adv. Math. 245, 625--689 (2013; Zbl 1291.14006)], have a shifted symplectic structure in the sense of Pantev-Toën-Vaquié-Vezzosi [\textit{T. Pantev} et al., Publ. Math., Inst. Hautes Étud. Sci. 117, 271--328 (2013; Zbl 1328.14027)].Quotients of triangulated categories and equivalences of Buchweitz, Orlov, and Amiot-Guo-Kellerhttps://zbmath.org/1472.180092021-11-25T18:46:10.358925Z"Iyama, Osamu"https://zbmath.org/authors/?q=ai:iyama.osamu"Yang, Dong"https://zbmath.org/authors/?q=ai:yang.dong.1Summary: We give a simple sufficient condition for a Verdier quotient \(\mathcal{T}/\mathcal{S}\) of a triangulated category \(\mathcal{T}\) by a thick subcategory \(\mathcal{S}\) to be realized inside of \(\mathcal{T}\) as an ideal quotient. As applications, we deduce three significant results by \textit{R.-O. Buchweitz} [``Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings'', Preprint, \url{https://hdl.handle.net/1807/16682}], \textit{D. Orlov} [Prog. Math. 270, 503--531 (2009; Zbl 1200.18007)] and \textit{C. Amiot} [Ann. Inst. Fourier 59, No. 6, 2525--2590 (2009; Zbl 1239.16011)], \textit{L. Guo} [J. Pure Appl. Algebra 215, No. 9, 2055--2071 (2011; Zbl 1239.16012)], \textit{B. Keller} [Doc. Math. 10, 551--581 (2005; Zbl 1086.18006); with \textit{M. Van den Bergh}, J. Reine Angew. Math. 654, 125--180 (2011; Zbl 1220.18012)].Euler characteristic of Springer fibershttps://zbmath.org/1472.200982021-11-25T18:46:10.358925Z"Kim, D."https://zbmath.org/authors/?q=ai:kim.dowon|kim.dohyeung|kim.dong-w|kim.do-young|kim.dong-yeon|kim.dong-il|kim.do-joong|kim.do-jin|kim.dojin|kim.dongkyu|kim.duksu|kim.dongkeon|kim.donghak|kim.doyeob|kim.donghan.1|kim.daejun|kim.dohwan|kim.doosoo|kim.dennis|kim.de-chan|kim.daniel|kim.dae-woo|kim.donghyun.1|kim.donghyun.2|kim.donghyun.3|kim.do-nyeon|kim.daeki|kim.dong-hwa|kim.donguk|kim.dai-young|kim.dae-sig|kim.dongsup|kim.dong-pyo|kim.dae-kyoo|kim.dong-heon|kim.dongsook|kim.dongwoo|kim.dukpa|kim.dae-nyeon|kim.dongjoo|kim.dohoon|kim.do-min|kim.do-woo|kim.daesoo|kim.daeju|kim.dong-hee|kim.do-kyung|kim.du-jin|kim.duk-sun|kim.dahn-goon|kim.doo-ho|kim.dai-gyoung|kim.dong-ah|kim.duck-hoon|kim.daehwan|kim.dongchoul|kim.du-won|kim.donggyu|kim.dae-eun|kim.dae-june|kim.daeyong|kim.deok-ho|kim.dalho|kim.david-s|kim.dongkyun|kim.dae-su|kim.dae-shik|kim.doo-hyun|kim.doyon|kim.dal-ho|kim.deokhwan|kim.donggun|kim.dok-yong|kim.dongryeol|kim.do-nyun|kim.donglok|kim.dongmok|kim.dong-yeol|kim.dongseok|kim.dong-in|kim.dong-seong|kim.do-sang|kim.dongku|kim.daekyu|kim.daejung|kim.duckkyung|kim.dae-woon|kim.dae-seoung|kim.dong-yun|kim.dong-sik|kim.dong-hyawn|kim.daeho|kim.dongun|kim.dongwon|kim.daesung|kim.duk-hyun|kim.dohyung|kim.dong-seon|kim.dongryul|kim.dongsu|kim.dookie|kim.doyoon|kim.daewa|kim.duckjin|kim.dokyoung|kim.daehong|kim.dae-hwang|kim.dongwook|kim.dongsoo-s|kim.duk-hun|kim.dongseung|kim.donggyun|kim.duk-gyoo|kim.dongsun|kim.dongju|kim.dano|kim.dmitry|kim.donghyok|kim.david-hong-kyun|kim.doheon|kim.do-gyun|kim.dong-kyoo|kim.daeryong|kim.dae-kyung|kim.dong-seo|kim.doyub|kim.do-jong|kim.daehak|kim.daegyoum|kim.dae-kwan|kim.denise|kim.djun-maximilian|kim.dong-guen|kim.dong-kie|kim.doochul|kim.doojin|kim.dongmin|kim.deok-jin|kim.dohan|kim.deokseong|kim.dae-mann|kim.daejoong|kim.dong-j|kim.dong-ok|kim.daeyoung|kim.david-b|kim.dean|kim.dongho|kim.doh-suk|kim.dohyeon|kim.daehyun|kim.daecheol|kim.dae-kyung.1|kim.dohyun|kim.dong-hyuk|kim.do-yeong|kim.dae-seung|kim.dokyun|kim.dehee|kim.duhyeong|kim.dongkwan|kim.dongjin|kim.dong-kyue|kim.deok-soo|kim.dukwon|kim.daehee|kim.daehoon|kim.du-yong|kim.dugyu|kim.deokjoo|kim.duk-kyung|kim.dae-hyeong|kim.dongkeun|kim.dongkook|kim.dohyeong|kim.donghoh|kim.donghwan|kim.daeyeoul|kim.doh-hyun|kim.dong-hoi|kim.dae-woong|kim.dongyung|kim.daeg-gun|kim.do-wan|kim.daewook|kim.dong-chan|kim.daijin|kim.daewon|kim.doe-wan|kim.dongcheol|kim.doo-seok|kim.dongjae|kim.dong-kwon|kim.daejin|kim.dong-chule|kim.dorian|kim.daehyon|kim.don-hee|kim.donghoonSummary: For Weyl groups of classical types, we present formulas to calculate the restriction of Springer representations to a maximal parabolic subgroup of the same type. As a result, we give recursive formulas for Euler characteristics of Springer fibers for classical types. We also give tables of those for exceptional types.Low degree cohomology of Frobenius kernelshttps://zbmath.org/1472.201042021-11-25T18:46:10.358925Z"Ngo, Nham V."https://zbmath.org/authors/?q=ai:ngo.nham-voSummary: Let \(G\) be a simple algebraic group defined over an algebraically closed field of characteristic \(p>0\). For a positive integer \(r\), let \(G_r\) be the \(r\)-th Frobenius kernel of \(G\). We determine in this paper a number \(m\) such that the cohomology \(\mathrm{H}^n(G_r,k)\) is isomorphic to \(\mathrm{H}^n(G_1,k)\) for all \(n\le m\) where \(m\) depends on \(p\) and the type of \(G\).
For the entire collection see [Zbl 1411.13002].The mapping class group action on \(\mathrm{SU}(3)\)-character varietieshttps://zbmath.org/1472.220132021-11-25T18:46:10.358925Z"Goldman, William M."https://zbmath.org/authors/?q=ai:goldman.william-m"Lawton, Sean"https://zbmath.org/authors/?q=ai:lawton.sean"Xia, Eugene Z."https://zbmath.org/authors/?q=ai:xia.eugene-zLet \(\Sigma\) be a compact oriented surface of genus \(g\) with boundary \(\partial \Sigma\), which has \(n \ge 1\) components. Let us consider the group of orientation-preserving homeomorphisms of \(\Sigma\) which fix \(\partial \Sigma\) pointwise. The mapping class group \(\Gamma\) of \(\Sigma\) is the group of connected components of this group of homeomorphisms. The authors also define for algebraic group \(G\) the relative character variety \(\mathcal M_{\mathcal C}(G)\).
W. Goldman conjectures many years ago that \(\Gamma\) acts ergodically on \(\mathcal M_{\mathcal C}(K)\) for compact Lie groups \(K\) and proved this conjecture for some compact Lie groups. Later some other results in this direction was proved. In this article the case \(K=\mathrm{SU}(3)\), \(g=1\) and \(n=1\) is considered. It is proved that aforementioned action of \(\Gamma\) in this case is ergodic with respect to the natural symplectic measure on the character variety.Global variants of Hartogs' theoremhttps://zbmath.org/1472.320022021-11-25T18:46:10.358925Z"Bochnak, Jacek"https://zbmath.org/authors/?q=ai:bochnak.jacek"Kucharz, Wojciech"https://zbmath.org/authors/?q=ai:kucharz.wojciechThis is a very interesting paper. The classical Hartogs theorem about separately analytic complex functions being analytic finds here a very fine generalization, which is also global. Let's have a look at the theorems proved in the paper:
Theorem 1.1. Let \(X=X_1\times\cdots\times X_n\) be the product of \(n\) complex algebraic manifolds and let \(f:U\to C\) be a function defined on an open subset \(U\) of \(X\). Assume that for each nonsingular algebraic curve \(C\subset X\), parallel to one of the factors of \(X\), the restriction \(f|U\cap C\) is a holomorphic function. Then \(f\) is a holomorphic function.
Theorem 1.2. Let \(X=X_1\times\cdots\times X_n\) be the product of \(n\) complex algebraic manifolds and let \(f:U\to C\) be a function defined on an open subset \(U\) of \(X\). Assume that for each nonsingular algebraic curve \(C\subset X\), parallel to one of the factors of \(X\), the restriction \(f|U\cap C\) is a Nash function. Then \(f\) is a Nash function.
Theorems 1.1 and 1.2 have a suitable analogue for regular functions.
Definition. Let \(X\) be a complex algebraic manifold. A function \(f:U\to C\), defined on an open subset \(U\) of \(X\), is said to be regular if there exists a rational function \(R\) on \(X\) such that \(U\subset X\smallsetminus\mathrm{Pole}(R)\) and \(f=R|U\), where Pole\((R)\) stands for the polar set of \(R\). Clearly, any regular function on \(U\) is a Nash function.
Theorem 1.3. Let \(X=X_1\times\cdots\times X_n\) be the product of \(n\) complex algebraic manifolds and let \(f:U\to C\) be a function defined on an open subset \(U\) of \(X\). Assume that for each nonsingular algebraic curve \(C\subset X\), parallel to one of the factors of \(X\), the restriction \(f|U\cap C\) is a regular function. Then \(f\) is a regular function.
The definition of a curve being ``parallel'' goes as follows: We say that a subset \(A\) of \(X\) is parallel to the \(i\)-th factor of \(X\) if \(\pi_j(A)\) consists of one point for each \(j=i\).
The paper is very well written, which is typical for these authors, very well organized, very clear, but the methods used are not easy.
For instance, in Proposition 2.3 below the Hironaka desingularization theorem is to be used. Not surprising, given the strength of the result, but worth noticing.
Proposition 2.3. Let \(X\) be a complex algebraic manifold and let \(f:U\to C\) be a function defined on an open subset \(U\) of \(X\). Assume that for each nonsingular algebraic curve \(C\subset X\) the restriction \(f|U\cap C\) is a holomorphic function. Then \(f\) is a holomorphic function.
The proof begins: According to Hironaka's theorem on resolution of singularities [\textit{H. Hironaka}, Ann. Math. (2) 79, 109--203 (1964; Zbl 0122.38603)], we may assume that the manifold \(X\) is projective. Hironaka's theorem cannot be avoided here.
The paper is virtually self contained, as the results used are well quoted and easy to find, even if difficult in themselves.
Each tool that needs adjusting (like Noether's normalization lemma presented here as Lemma 1.2) is adjusted and a full proof is given. This makes the paper very pleasant for the reader.
The results seem very useful. It is often easier to examine functions on algebraic curves (cf. [\textit{J. Kollár} et al., Math. Ann. 370, No. 1--2, 39--69 (2018; Zbl 1407.14056)]).Orthogonality of divisorial Zariski decompositions for classes with volume zerohttps://zbmath.org/1472.320092021-11-25T18:46:10.358925Z"Tosatti, Valentino"https://zbmath.org/authors/?q=ai:tosatti.valentinoConsider the following statement:
Conjecture: Let \((X, \omega)\) be a compact Kähler manifold, and \(\alpha\) a pseudoeffective \((1,1)\) class. Then \[ \langle \alpha^{n-1} \rangle \cdot \alpha = \mathrm{Vol}(\alpha), \] where \(\textrm{Vol}(\alpha)\) is the volume of the class \(\alpha\) and \(\langle \cdot \rangle\) is the moving intersection product of classes in the sense of Boucksom.
The above is known as the orthogonality conjecture for divisorial Zariski decompositions, which was observed by \textit{S. Boucksom} et al. [J. Algebr. Geom. 22, No. 2, 201--248 (2013; Zbl 1267.32017); J. Algebr. Geom. 18, No. 2, 279--308 (2009; Zbl 1162.14003)] and is equivalent to the weak transcendental Morse inequalities, the \(C^1\) differentiability of the volume function on the big cone, and the ``cone duality'' conjecture, i.e., \textit{the dual cone of the pseudoeffective cone is the movable cone}.
This was proven for \(X\) projective in [\textit{S. Boucksom} et al., J. Algebr. Geom. 22, No. 2, 201--248 (2013; Zbl 1267.32017); \textit{D. W. Nyström}, J. Am. Math. Soc. 32, No. 3, 675--689 (2019; Zbl 1429.32031)], and formulated as a conjecture on arbitrary compact Kähler manifolds in [\textit{S. Boucksom} et al., J. Algebr. Geom. 22, No. 2, 201--248 (2013; Zbl 1267.32017)]. The main result of this note is a proof of the orthogonality conjecture on arbitrary compact Kähler manifolds for pseudoeffective \((1,1)\) classes that are assumed to have volume zero.Computing regular meromorphic differential forms via Saito's logarithmic residueshttps://zbmath.org/1472.320132021-11-25T18:46:10.358925Z"Tajima, Shinichi"https://zbmath.org/authors/?q=ai:tajima.shinichi"Nabeshima, Katsusuke"https://zbmath.org/authors/?q=ai:nabeshima.katsusukeThe concept of regular meromorphic differential forms was introduced independently by \textit{E. Kunz} [Manuscr. Math. 15, 91--108 (1975; Zbl 0299.14013)] and \textit{D. Barlet} [C. R. Acad. Sci., Paris, Sér. A 282, 579--582 (1976; Zbl 0323.32006)]. Somewhat later the reviewer [Adv. Sov. Math. 1, 211--246 (1990; Zbl 0731.32005)] proved that in the hypersurface case such forms naturally appear as the image of the logarithmic residue introduced by \textit{K. Saito} [J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 265--291 (1980; Zbl 0496.32007)]. The authors of the paper under review present an algorithm for computing regular meromorphic differential forms using Saito's residue and torsion differentials of the modules of regular holomorphic forms (see [the rewiever, Complex Variables, Theory Appl. 50, No. 7--11, 777--802 (2005; Zbl 1083.32024)]. Then their constructions are applied for explicit computations of the Gauss-Manin connection in the case of isolated hypersurface singularities. As an example, the cases of functions of two and three variables are analyzed in detail.Adiabatic limit and the Frölicher spectral sequencehttps://zbmath.org/1472.320192021-11-25T18:46:10.358925Z"Popovici, Dan"https://zbmath.org/authors/?q=ai:popovici.danIn complex geometry, it is well known that the Frölicher spectral sequence of a compact Kähler manifold degenerates at \(E_1\) page (in particular it degenerates at \(E_2\) page). Since the Kähler condition is quite restrictive for compact complex manifolds of dimension at least 3, it is natural to seek other metric conditions which ensure the \(E_2\)-degeneration of the Frölicher spectral sequence.
Let \(X\) be a compact complex manifold with a Hermitian metric \(\omega\). In this article, the author gives a sufficient metric condition for degeneration at \(E_2\), which roughly says that the torsion of \(\omega\) is ``small''. One of the new ideas is to consider the rescalings of \(\omega\) and \(\partial\), which is an adaption of the adiabatic limit construction associated with a Riemann foliation (see, e.g., [\textit{E. Witten}, Commun. Math. Phys. 100, 197--229 (1985; Zbl 0581.58038)]) to the case of the splitting \(d=\partial +\overline{\partial}\). It seems interesting to point out that similar ideas also appeared in the setting of non-abelian Hodge theory, see [\textit{C. Simpson}, Mixed twistor structures, arXiv preprint alg-geom/9705006, 1997] and Theorem 2.2.4 in [\textit{C. Sabbah}, Polarizable twistor \(\mathcal{D}\)-modules. Paris: Sociéteé Mathématique de France (2005; Zbl 1085.32014)].
Moreover, using a variant of the Efremov-Shubin variational principle, along with the pesudodifferential Laplacian in [the author, Int. J. Math. 27, No. 14, Article ID 1650111, 31 p. (2016; Zbl 1365.53067)] and Demaily's Bochner-Kodaira-Nakano formula for Hermitian metrics, the author finds a formula for the dimensions of the vector spaces on each page of the Frölicher spectral sequence in terms of of the number of small eigenvalues of the rescaled Laplacian. This formula is of independent interest, and is inspired by the analogous result for foliations proven in [\textit{J. A. Álvarez López} and \textit{Y. A. Kordyukov}, Geom. Funct. Anal. 10, No. 5, 977--1027 (2000; Zbl 0965.57024)].The algebraic theory of fractional jumpshttps://zbmath.org/1472.370922021-11-25T18:46:10.358925Z"Goldfeld, Dorian"https://zbmath.org/authors/?q=ai:goldfeld.dorian-m"Micheli, Giacomo"https://zbmath.org/authors/?q=ai:micheli.giacomoSummary: In this paper, we start by briefly surveying the theory of fractional jumps and transitive projective maps. Then we show some new results on the absolute jump index, on projectively primitive polynomials, and on compound constructions.
For the entire collection see [Zbl 1455.11009].On semigroup orbits of polynomials and multiplicative ordershttps://zbmath.org/1472.370932021-11-25T18:46:10.358925Z"Mello, Jorge"https://zbmath.org/authors/?q=ai:mello.jorgeSummary: We show, under some natural restrictions, that some semigroup orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime \(p\), extending previous work of \textit{I. E. Shparlinski} [Glasg. Math. J. 60, No. 2, 487--493 (2018; Zbl 1431.11130)].The complete classification of empty lattice 4-simpliceshttps://zbmath.org/1472.520172021-11-25T18:46:10.358925Z"Iglesias-Valiño, Óscar"https://zbmath.org/authors/?q=ai:iglesias-valino.oscar"Santos, Francisco"https://zbmath.org/authors/?q=ai:santos.franciscoSummary: An empty simplex is a lattice simplex with only its vertices as lattice points. Their classification in dimension three was completed by G. White in 1964. In 1988, S. Mori, D. R. Morrison, and I. Morrison started the task in dimension four, with their motivation coming from the close relationship between empty simplices and terminal quotient singularities. They conjectured a classification of empty simplices of prime volume, modulo finitely many exceptions. Their conjecture was proved by Sankaran (1990) with a simplified proof by Bober (2009). The same classification was claimed by Barile et al. in 2011 for simplices of non-prime volume, but this statement was proved wrong by Blanco et al. (2016).
In this article, we complete the classification of 4-dimensional empty simplices. In doing so, we correct and complete the classification by Barile et al., and we also compute all the finitely many exceptions, by first proving an upper bound for their volume. The whole classification has:
\begin{itemize}
\item[1)] One 3-parameter family, consisting of simplices of width equal to one.
\item[2)] Two 2-parameter families (the one in Mori et al., plus a second new one).
\item[3)] Forty-six 1-parameter families (the 29 in Mori et al., plus 17 new ones).
\item[4)] 2461 individual simplices not belonging to the above families, with (normalized) volumes ranging between 24 and 419.
\end{itemize}
We characterize the infinite families of empty simplices in terms of the lower dimensional point configurations that they project to, with techniques that can potentially be applied to higher dimensions and other classes of lattice polytopes.Multi-splits and tropical linear spaces from nested matroidshttps://zbmath.org/1472.520192021-11-25T18:46:10.358925Z"Schröter, Benjamin"https://zbmath.org/authors/?q=ai:schroter.benjaminSummary: We present an explicit combinatorial description of a special class of facets of the secondary polytopes of hypersimplices. These facets correspond to polytopal subdivisions called multi-splits. We show that the maximal cells in a multi-split of a hypersimplex are matroid polytopes of nested matroids. Moreover, we derive a description of all multi-splits of a product of simplices. Additionally, we present a computational result to derive explicit lower bounds on the number of facets of secondary polytopes of hypersimplices.A moment map interpretation of the Ricci form, Kähler-Einstein structures, and Teichmüller spaceshttps://zbmath.org/1472.530922021-11-25T18:46:10.358925Z"García-Prada, Oscar"https://zbmath.org/authors/?q=ai:garcia-prada.oscar"Salamon, Dietmar"https://zbmath.org/authors/?q=ai:salamon.dietmar-aSummary: This paper surveys the role of moment maps in Kähler geometry. The first section discusses the Ricci form as a moment map and then moves on to moment map interpretations of the Kähler-Einstein condition and the scalar curvature (Quillen-Fujiki-Donaldson). The second section examines the ramifications of these results for various Teichmüller spaces and their Weil-Petersson symplectic forms and explains how these arise naturally from the construction of symplectic quotients. The third section discusses a symplectic form introduced by Donaldson on the space of Fano complex structures.
For the entire collection see [Zbl 1461.37002].Stability of the conical Kähler-Ricci flows on Fano manifoldshttps://zbmath.org/1472.531062021-11-25T18:46:10.358925Z"Liu, Jiawei"https://zbmath.org/authors/?q=ai:liu.jiawei"Zhang, Xi"https://zbmath.org/authors/?q=ai:zhang.xiSummary: In this paper, we study stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle \(2\pi\beta\) along the divisor, then for any \(\beta'\) sufficiently close to \(\beta\), the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with cone angle \(2\pi\beta'\) along the divisor. Here, we only use the condition that the Log Mabuchi energy is bounded from below. This is a weaker condition than the properness that we have adopted to study the convergence. As applications, we give parabolic proofs of Donaldson's openness theorem and his conjecture for the existence of conical Kähler-Einstein metrics with positive Ricci curvatures.Riccati-type pseudo-potentials, conservation laws and solitons of deformed sine-Gordon modelshttps://zbmath.org/1472.810672021-11-25T18:46:10.358925Z"Blas, H."https://zbmath.org/authors/?q=ai:blas.harold"Callisaya, H. F."https://zbmath.org/authors/?q=ai:callisaya.hector-flores"Campos, J. P. R."https://zbmath.org/authors/?q=ai:campos.j-p-rSummary: Deformed sine-Gordon (DSG) models \(\partial_\xi \partial_\eta w + \frac{d}{d w} V(w) = 0\), with \(V(w)\) being the deformed potential, are considered in the context of the Riccati-type pseudo-potential approach. A compatibility condition of the deformed system of Riccati-type equations reproduces the equation of motion of the DSG models. Then, we provide a pair of linear systems of equations for the DSG model and an associated infinite tower of non-local conservation laws. Through a direct construction and supported by numerical simulations of soliton scatterings, we show that the DSG models, which have recently been defined as quasi-integrable in the anomalous zero-curvature approach [\textit{L. A. Ferreira} and \textit{W. J. Zakrzewski}, J. High Energy Phys. 2011, No. 5, Paper No. 130, 39 p. (2011; Zbl 1296.81035)], possess new towers of infinite number of quasi-conservation laws. We compute numerically the first sets of non-trivial and independent charges (beyond energy and momentum) of the DSG model: the two third order conserved charges and the two fifth order asymptotically conserved charges in thepseudo-potential approach, and the first four anomalies of the new towers of charges, resp ectively. We consider kink-kink, kink-antikink and breather configurations for the Bazeia et al. potential \(V_q(w) = \frac{64}{q^2} \tan^2 \frac{w}{2}(1 - | \sin \frac{w}{2} |^q)^2\) \((q \in \mathbb{R})\), which contains the usual SG potential \(V_2(w) = 2 [1 - \cos(2w)]\). The numerical simulations are performed using the 4th order Runge-Kutta method supplied with non-reflecting boundary conditions.Four-dimensional Fano quiver flag zero locihttps://zbmath.org/1472.810772021-11-25T18:46:10.358925Z"Kalashnikov, Elana"https://zbmath.org/authors/?q=ai:kalashnikov.elanaSummary: Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of representation theoretic vector bundles. We prove the Abelian/non-Abelian correspondence in this context: this allows us to compute genus zero Gromov-Witten invariants of quiver flag zero loci. We determine the ample cone of a quiver flag variety, and disprove a conjecture of Craw. In the appendices (which can be found in the electronic supplementary material), which are joint work with Tom Coates and Alexander Kasprzyk, we use these results to find four-dimensional Fano manifolds that occur as quiver flag zero loci in ambient spaces of dimension up to 8, and compute their quantum periods. In this way, we find at least 141 new four-dimensional Fano manifolds.Gauge transformations of spectral triples with twisted real structureshttps://zbmath.org/1472.810822021-11-25T18:46:10.358925Z"Magee, Adam M."https://zbmath.org/authors/?q=ai:magee.adam-m"Dąbrowski, Ludwik"https://zbmath.org/authors/?q=ai:dabrowski.ludwikSummary: Twisted real structures are well-motivated as a way to implement the conformal transformation of a Dirac operator for a real spectral triple without needing to twist the noncommutative one-forms. We study the coupling of spectral triples with twisted real structures to gauge fields, adopting Morita equivalence via modules and bimodules as a guiding principle and paying special attention to modifications to the inner fluctuations of the Dirac operator. In particular, we analyze the twisted first-order condition as a possible alternative to abandoning the first-order condition in order to go beyond the standard model and elaborate upon the special case of gauge transformations accordingly. Applying the formalism to a toy model, we argue that under certain physically motivated assumptions, the spectral triple based on the left-right symmetric algebra should reduce to that of the standard model of fundamental particles and interactions, as in the untwisted case.
{\copyright 2021 American Institute of Physics}A new spectral analysis of stationary random Schrödinger operatorshttps://zbmath.org/1472.810922021-11-25T18:46:10.358925Z"Duerinckx, Mitia"https://zbmath.org/authors/?q=ai:duerinckx.mitia"Shirley, Christopher"https://zbmath.org/authors/?q=ai:shirley.christopherThe authors consider random Schrödinger operators of the form
\[
-\Delta + \lambda V_{\omega}
\]
and the associated Schrödinger equation, where \(V_{\omega}\) is a realization of a stationary random potential \(V\). The regime under consideration here is \(0<\lambda \ll 1\). The main goal of the authors is to develop a spectral approach to describe the long time behavior of the system beyond perturbative timescales by using ideas from Malliavin calculus, leading to rigorous Mourre type results. In particular, the authors describe the dynamics by a fibered family of spectral perturbation problems. They then state a number of exact resonance conjectures which would require that Bloch waves exist as resonant modes. An approximate resonance result is obtained and the first spectral proof of the decay of time correlations on the kinetic timescale is also provided.Spinorial \(R\) operator and algebraic Bethe ansatzhttps://zbmath.org/1472.811202021-11-25T18:46:10.358925Z"Karakhanyan, D."https://zbmath.org/authors/?q=ai:karakhanyan.david"Kirschner, R."https://zbmath.org/authors/?q=ai:kirschner.rolandSummary: We propose a new approach to the spinor-spinor R-matrix with orthogonal and symplectic symmetry. Based on this approach and the fusion method we relate the spinor-vector and vector-vector monodromy matrices for quantum spin chains. We consider the explicit spinor \(R\) matrices of low rank orthogonal algebras and the corresponding \(RTT\) algebras. Coincidences with fundamental \(R\) matrices allow to relate the Algebraic Bethe Ansatz for spinor and vector monodromy matrices.Field theory and \(\lambda\)-deformations: expanding around the identityhttps://zbmath.org/1472.811442021-11-25T18:46:10.358925Z"Georgiou, George"https://zbmath.org/authors/?q=ai:georgiou.george-m"Sfetsos, Konstantinos"https://zbmath.org/authors/?q=ai:sfetsos.konstantinosSummary: We explore the structure of the \(\lambda\)-deformed \(\sigma\)-model action by setting up a perturbative expansion around the free field point corresponding to the identity group element. We include all field interaction terms up to sixth order. We compute the two- and three-point functions of current and primary field operators, their anomalous dimensions as well as the \(\beta\)-function for the \(\lambda\)-parameter. Our results are in complete agreement with those obtained previously using gravitational and/or CFT perturbative methods in conjunction with the non-perturbative symmetry, as well as with those obtained using methods exploiting the geometry defined in the space of couplings. The advantage of this approach is that all deformation effects are already encoded in the couplings of the interaction vertices and in the \(\lambda\)-dressed operators.Mocking the \(u\)-plane integralhttps://zbmath.org/1472.811842021-11-25T18:46:10.358925Z"Korpas, Georgios"https://zbmath.org/authors/?q=ai:korpas.georgios"Manschot, Jan"https://zbmath.org/authors/?q=ai:manschot.jan"Moore, Gregory W."https://zbmath.org/authors/?q=ai:moore.gregory-w"Nidaiev, Iurii"https://zbmath.org/authors/?q=ai:nidaiev.iuriiThe authors consider the topologically twisted counterpart of \(\mathcal{N} = 2\) supersymmetric Yang-Mills theory with gauge group SU(2) in the presence of arbitrary 't~Hooft flux [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. The gauge group is broken to U(1) on the Coulomb branch. The Coulomb branch, also known as the \(u\)-plane, can be considered as a three-punctured sphere, where the punctures correspond to the weak coupling limit, and the two strong coupling singularities.
The authors consider compact four-manifolds \(M\) with \((b_1, b^+_2 ) = (0, 1)\) and without boundary. Here \(b_j\) denotes the Betti numbers of \(M\), and \(b^+_2\) is the number of positive definite eigenvalues of the intersection form of two-cycles of \(M\). Complex four-manifolds with \(b^+_2= 1\) are well-studied and classified by the Enriques-Kodaira classification reviewed in section 3.2 of this paper.
The authors evaluate and analyze the \(u\)-plane contribution to the correlation functions known as the \(u\)-plane integral and show that they can be evaluated by integration by parts leading to expressions in terms of mock modular forms for point observables, and Appell-Lerch sums for surface observables.
In the absence of hypermultiplets, we are always free to consider the case where the principal SO(3) gauge bundle has a nontrivial 't~Hooft flux \(w_2\in H^2(M;\mathbb{Z}_2)\). The authors choose an integral lift \(\overline{w_2}\) (which is supposed to exist) such that \(\mu:=\frac{1}{2}\overline{w_2}\in H^2(M;\mathbb{R})\). The path integral over the Coulomb branch of Donaldson-Witten theory, denoted by \(\Phi^J_\mu\) is an integral over the infinite dimensional field space, which reduces to a finite dimensional integral over the zero modes [\textit{G. Moore} and \textit{E. Witten}, Adv. Theor. Math. Phys. 1, No. 2, 298--387 (1997; Zbl 0899.57021)]. \(J\in H^2(M,\mathbb{R})\) is the period point which depends on the metric due to the self-duality condition.
The explicit expressions for the \(u\)-plane integrals are given in equation (5.44) for manifolds with odd intersection form and just point observables inserted, equation (5.65) for manifolds with odd intersection form and just surface observables inserted, and equation (5.84) for manifolds with even intersection form and just surface observables inserted. These expressions hold for a special choice of the metric, though the metric dependence only enters through the choice of period point \(J\). Using the expression for the wall-crossing formula in terms of indefinite theta functions, analogous mock modular forms relevant to other chambers can be obtained [\textit{L. Göttsche} and \textit{D. Zagier}, Sel. Math., New Ser. 4, No. 1, 69--115 (1998; Zbl 0924.57025)].
Using the expression for \(\Phi^J_\mu[\mathcal{O}]\) in terms of mock modular forms, one can address analytic properties of the correlators for \(b^+_2= 1\), analogously to the structural results for manifolds with \(b^+_2 > 1\) [\textit{P. B. Kronheimer} and \textit{T. S. Mrowka}, J. Differ. Geom. 41, No. 3, 573--734 (1995; Zbl 0842.57022)]. The authors study the asymptotic behavior of \(\Phi[u^\ell]\) for large \(\ell\) and find experimental evidence that \(\Phi^J_\mu[u^\ell]\sim 1/(\ell\log(\ell))\) for any four-manifold with \((b_1, b^+_2 ) = (0, 1)\). The asymptotic behavior of \(\Phi^J_\mu[u^\ell]\) suggests that
\[
\Phi^J_\mu[e^{2pu}]=\sum_{\ell\ge0}(2p)^\ell\Phi^J_\mu[u^\ell]/\ell!
\]
is an entire function of \(p\) rather than a formal expansion. The authors find similar experimental evidence for the \(u\)-plane contribution to the exponentiated surface observable.Closed superstring moduli tree-level two-point scattering amplitudes in type IIB orientifold on \(T^6/(Z_2 \times Z_2)\)https://zbmath.org/1472.811892021-11-25T18:46:10.358925Z"Aldi, Alice"https://zbmath.org/authors/?q=ai:aldi.alice"Firrotta, Maurizio"https://zbmath.org/authors/?q=ai:firrotta.maurizioSummary: We reconsider the two-point string scattering amplitudes of massless Neveu-Schwarz-Neveu-Schwarz states of Type IIB orientifold superstring theory on the disk and projective plane in ten dimensions and analyse its \(\alpha^\prime\) expansion. We also discuss the unoriented Type IIB theory on \(T^6 / \mathbb{Z}_2 \times \mathbb{Z}_2\) where two-point string scattering amplitudes of the complex Kähler moduli and complex structures of the untwisted sector are computed on the disk and projective plane. New results are obtained together with known ones. Finally, we compare string scattering amplitudes results at \({\alpha^{\prime}}^2\)-order with the (curvature)\(^2\) terms in the low energy effective action of D-branes and \(\Omega \)-planes in both cases.Entropic \(c\)-functions in \(T \overline{T}\), \(J \overline{T}\), \(T \overline{J}\) deformationshttps://zbmath.org/1472.811902021-11-25T18:46:10.358925Z"Asrat, Meseret"https://zbmath.org/authors/?q=ai:asrat.meseretSummary: We study the holographic entanglement entropy of an interval in a quantum field theory obtained by deforming a holographic two-dimensional conformal field theory via a general linear combination of irrelevant operators that are closely related to, but nonetheless distinct from, \(T \overline{T}\), \(J \overline{T}\) and \(T \overline{J} \), and compute the Casin-Huerta entropic \(c\)-function. In the ultraviolet, for a particular combination of the deformation parameters, we find that the leading order dependence of the entanglement entropy on the length of the interval is given by a square root but not logarithmic term. Such power law dependence of the entanglement entropy on the interval length is quite peculiar and interesting. We also find that the entropic \(c\)-function is ultraviolet regulator independent, and along the renormalization group upflow towards the ultraviolet, it is non-decreasing. We show that in the ultraviolet the entropic \(c\)-function exhibits a power law divergence as the interval length approaches a minimum finite value determined in terms of the deformation parameters. This value sets the non-locality scale of the theory.Quantization of Harer-Zagier formulashttps://zbmath.org/1472.811982021-11-25T18:46:10.358925Z"Morozov, A."https://zbmath.org/authors/?q=ai:morozov.albert-dmitrievich|morozov.a-c|morozov.a-m|morozov.andrei-alekseevich|morozov.a-v|morozov.a-k|morozov.alexander-yu|morozov.a-a|morozov.a-g|morozov.anton|morozov.alexandre-v|morozov.andrew-yu|morozov.alexander-n|morozov.andrey-n|morozov.alexei-yurievich|morozov.andrei-sergeevich"Popolitov, A."https://zbmath.org/authors/?q=ai:popolitov.aleksandr|popolitov.a-v"Shakirov, Sh."https://zbmath.org/authors/?q=ai:shakirov.shamil|shakirov.sh-rSummary: We derive the analogues of the Harer-Zagier formulas for single- and double-trace correlators in the q-deformed Hermitian Gaussian matrix model. This fully describes single-trace correlators and opens a road to \(q\)-deformations of important matrix models properties, such as genus expansion and Wick theorem.Deformations, renormgroup, symmetries, AdS/CFThttps://zbmath.org/1472.812102021-11-25T18:46:10.358925Z"Mikhailov, Andrei"https://zbmath.org/authors/?q=ai:mikhailov.andrei-yu|mikhailov.andrei-igorevichSummary: We consider the deformations of a supersymmetric quantum field theory by adding spacetime-dependent terms to the action. We propose to describe the renormalization of such deformations in terms of some cohomological invariants, a class of solutions of a Maurer-Cartan equation. We consider the strongly coupled limit of \(N = 4\) supersymmetric Yang-Mills theory. In the context of AdS/CFT correspondence, we explain what corresponds to our invariants in classical supergravity. There is a leg amputation procedure, which constructs a solution of the Maurer-Cartan equation from tree diagrams of SUGRA. We consider a particular example of the beta-deformation. It is known that the leading term of the beta-function is cubic in the parameter of the beta-deformation. We give a cohomological interpretation of this leading term. We conjecture that it is actually encoded in some simpler cohomology class, which is quadratic in the parameter of the beta-deformation.\(O(d,d)\) transformations preserve classical integrabilityhttps://zbmath.org/1472.812272021-11-25T18:46:10.358925Z"Orlando, Domenico"https://zbmath.org/authors/?q=ai:orlando.domenico"Reffert, Susanne"https://zbmath.org/authors/?q=ai:reffert.susanne"Sekiguchi, Yuta"https://zbmath.org/authors/?q=ai:sekiguchi.yuta"Yoshida, Kentaroh"https://zbmath.org/authors/?q=ai:yoshida.kentarohSummary: In this note, we study the action of \(O(d, d)\) transformations on the integrable structure of two-dimensional non-linear sigma models via the doubled formalism. We construct the Lax pairs associated with the \(O(d, d)\)-transformed model and find that they are in general non-local because they depend on the winding modes. We conclude that every \(O(d, d; \mathbb{R})\) deformation preserves integrability. As an application we compute the Lax pairs for continuous families of deformations, such as \(J \bar{J}\) marginal deformations and TsT transformations of the three-sphere with \(H\)-flux.Time-space noncommutativity and Casimir effecthttps://zbmath.org/1472.812332021-11-25T18:46:10.358925Z"Harikumar, E."https://zbmath.org/authors/?q=ai:harikumar.e"Panja, Suman Kumar"https://zbmath.org/authors/?q=ai:panja.suman-kumar"Rajagopal, Vishnu"https://zbmath.org/authors/?q=ai:rajagopal.vishnuSummary: We show that the Casimir force and energy are modified in the \(\kappa\)-deformed space-time. This is shown by solving the Green's function corresponding to \(\kappa \)-deformed scalar field equation in presence of two parallel plates, modelled by \(\delta\)-function potentials. Exploiting the relation between Energy-Momentum tensor and Green's function, we calculate corrections to Casimir force, valid up to second order in the deformation parameter. The Casimir force is shown to get corrections which scale as \(L^{- 4}\) and \(L^{- 6}\) and both these types of corrections produce attractive forces. Using the measured value of Casimir force, we show that the deformation parameter should be below \(10^{-23}\) m.Defects, nested instantons and comet-shaped quivershttps://zbmath.org/1472.812362021-11-25T18:46:10.358925Z"Bonelli, G."https://zbmath.org/authors/?q=ai:bonelli.giulio"Fasola, N."https://zbmath.org/authors/?q=ai:fasola.nadir"Tanzini, A."https://zbmath.org/authors/?q=ai:tanzini.alessandroIn four-dimensional supersymmetric gauge theories, surface defects are real codimension two submanifolds where a specific reduction of the gauge connection takes place. They were pioneered by \textit{G. 't Hooft} [Nucl. Phys. B 153, 141--160 (1979; \url{doi:10.1016/0550-3213(79)90595-9})] in the classification of the phases of gauge theories and introduced in mathematics by \textit{P. B. Kronheimer} and \textit{T. S. Mrowka} [Topology 34, No. 1, 37--97 (1995; Zbl 0832.57011)] in the study of Donaldson invariants.
In this paper, the authors introduce and study surface defects supporting nested instantons with respect to the parabolic reduction of the gauge group at the defect. These defects are engineered from a D7/D3 brane system on a local compact complex surface. Specifically, they consider the product \(S=T^2\times C\) of a 2-torus \(T^2\) and a Riemann surface \(C\) with punctures \(\{p_i\}\) and embed it into CY 5-fold \(T^2\times T^*C\times \mathbb{C}^2\). Here surface operators are located at \(T^2\times \{p_i\}\). Supersymmetric partition functions of these systems provide conjectural formulae for virtual invariants of the moduli spaces. Mathematically, they obtain for a single D7-brane conjectural explicit formulae for the virtual equivariant elliptic genus of a certain bundle over the moduli space of the nested Hilbert scheme of points on the affine plane. Connections to Vafa-Witten theory and Donaldson-Thomas theory of CY 4-folds are also mentioned.Nilpotence varietieshttps://zbmath.org/1472.812382021-11-25T18:46:10.358925Z"Eager, Richard"https://zbmath.org/authors/?q=ai:eager.richard"Saberi, Ingmar"https://zbmath.org/authors/?q=ai:saberi.ingmar-a"Walcher, Johannes"https://zbmath.org/authors/?q=ai:walcher.johannesAlgebraic varieties coming from orbits of nilpotent elements of Lie algebras is an important subject, which has also recently found deep connections with theoretical physics. The current paper is a wonderful monograph on this subject, focusing on varieties canonically associated with any Lie superalgebra, and treating them from a systematic and new perspective of natural moduli spaces parameterizing twists of a super-Poincare-invariant physical theory.Towards an axiomatic formulation of noncommutative quantum field theory. II.https://zbmath.org/1472.812472021-11-25T18:46:10.358925Z"Chaichian, M."https://zbmath.org/authors/?q=ai:chaichian.masud"Mnatsakanova, M. N."https://zbmath.org/authors/?q=ai:mnatsakanova.m-n"Vernov, Yu. S."https://zbmath.org/authors/?q=ai:vernov.yu-sSummary: Classical results of the axiomatic quantum field theory -- irreducibility of the set of field operators, Reeh and Schlieder's theorems and generalized Haag's theorem are proven in \(SO(1, 1)\) invariant quantum field theory, of which an important example is noncommutative quantum field theory. In \(SO(1, 3)\) invariant theory new consequences of generalized Haag's theorem are obtained. It has been proven that the equality of four-point Wightman functions in two theories leads to the equality of elastic scattering amplitudes and thus the total cross-sections in these theories.
For Part I, see [the first author et al., J. Math. Phys. 52, No. 3, 032303, 13 p. (2011; Zbl 1315.81096)].Refined scattering diagrams and theta functions from asymptotic analysis of Maurer-Cartan equationshttps://zbmath.org/1472.812492021-11-25T18:46:10.358925Z"Leung, Naichung Conan"https://zbmath.org/authors/?q=ai:leung.naichung-conan"Ma, Ziming Nikolas"https://zbmath.org/authors/?q=ai:ma.ziming-nikolas"Young, Matthew B."https://zbmath.org/authors/?q=ai:young.matthew-bThe topic of the article arises to the reconstruction problem in [\textit{A. Strominger} et al., Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)] mirror symmetry. Previously the notion of a scattering diagramm was investigated by [\textit{M. Kontsevich} and \textit{Y. Soibelman}, Prog. Math. 244, 321--385 (2006; Zbl 1114.14027)]. The authors develop an asymptotic analytic approach to the study of scattering diagrams. They investigate the asymptotic behavior of Maurer-Cartan elements of a (dg) Lie algebra. The authors found an alternative geometric differential approach to the proofs of the consistent completion of the scattering diagrams, previously investigated by Kontsevich-Soibelman [loc. cit.], \textit{M. Gross} and \textit{B. Siebert} [Ann. Math. (2) 174, No. 3, 1301--1428 (2011; Zbl 1266.53074)] and \textit{T. Bridgeland} [Algebr. Geom. 4, No. 5, 523--561 (2017; Zbl 1388.16013)]. The paper under reviewing deals with the geometric interpretation of theta-functions, their wall-crossing, which allow to give a combinatorial description of Hall algebra theta functions for acyclic quivers with nondegenerate skew- symmetric Euler functionsDynamical Majorana neutrino masses and axions. I.https://zbmath.org/1472.812812021-11-25T18:46:10.358925Z"Alexandre, Jean"https://zbmath.org/authors/?q=ai:alexandre.jean"Mavromatos, Nick E."https://zbmath.org/authors/?q=ai:mavromatos.nick-e"Soto, Alex"https://zbmath.org/authors/?q=ai:soto.alexSummary: We discuss dynamical mass generation for fermions and (string-inspired) pseudoscalar fields (axion-like particles (ALP)), in the context of effective theories containing Yukawa type interactions between the fermions and ALPs. We discuss both Hermitian and non-Hermitian Yukawa interactions, which are motivated in the context of some scenarios for radiative (anomalous) Majorana sterile neutrino masses in some effective field theories. The latter contain shift-symmetry breaking Yukawa interactions between sterile neutrinos and ALPs. Our models serve as prototypes for discussing dynamical mass generation in string-inspired theories, where ALP potentials are absent in any order of string perturbation theory, and could only be generated by stringy non perturbative effects, such as instantons. The latter are also thought of as being responsible for generating our Yukawa interactions, which are thus characterised by very small couplings. We show that, for a Hermitian Yukawa interaction, there is no (pseudo)scalar dynamical mass generation, but there is fermion dynamical mass generation, provided one adds a bare (pseudo)scalar mass. The situation is opposite for an anti-Hermitian Yukawa model: there is (pseudo)scalar dynamical mass generation, but no fermion dynamical mass generation. In the presence of additional attractive four-fermion interactions, dynamical fermion mass generation can occur in these models, under appropriate conditions and range of their couplings.One residue to rule them all: electroweak symmetry breaking, inflation and field-space geometryhttps://zbmath.org/1472.812892021-11-25T18:46:10.358925Z"Karananas, Georgios K."https://zbmath.org/authors/?q=ai:karananas.georgios-k"Michel, Marco"https://zbmath.org/authors/?q=ai:michel.marco"Rubio, Javier"https://zbmath.org/authors/?q=ai:rubio.javierSummary: We point out that the successful generation of the electroweak scale via gravitational instanton configurations in certain scalar-tensor theories can be viewed as the aftermath of a simple requirement: the existence of a quadratic pole with a sufficiently small residue in the Einstein-frame kinetic term for the Higgs field. In some cases, the inflationary dynamics may also be controlled by this residue and therefore related to the Fermi-to-Planck mass ratio, up to possible uncertainties associated with the instanton regularization. We present here a unified framework for this hierarchy generation mechanism, showing that the aforementioned residue can be associated with the curvature of the Einstein-frame target manifold in models displaying spontaneous breaking of dilatations. Our findings are illustrated through examples previously considered in the literature.Two anyons on the sphere: nonlinear states and spectrumhttps://zbmath.org/1472.813072021-11-25T18:46:10.358925Z"Polychronakos, Alexios P."https://zbmath.org/authors/?q=ai:polychronakos.alexios-p"Ouvry, Stéphane"https://zbmath.org/authors/?q=ai:ouvry.stephaneSummary: We study the energy spectrum of two anyons on the sphere in a constant magnetic field. Making use of rotational invariance we reduce the energy eigenvalue equation to a system of linear differential equations for functions of a single variable, a reduction analogous to separating center of mass and relative coordinates on the plane. We solve these equations by a generalization of the Frobenius method and derive numerical results for the energies of non-analytically derivable states.Multitwistor mechanics of massless superparticle on \(AdS_5 \times S^5\) superbackgroundhttps://zbmath.org/1472.830192021-11-25T18:46:10.358925Z"Uvarov, D. V."https://zbmath.org/authors/?q=ai:uvarov.dmitriy-vSummary: Supertwistors relevant to \(AdS_5 \times S^5\) superbackground of IIB supergravity are studied in the framework of the \(D = 10\) massless superparticle model in the first-order formulation. Product structure of the background suggests using \(D = 1 + 4\) Lorentz-harmonic variables to express momentum components tangent to \(A d S_5\) and \(D = 5\) harmonics to express momentum components tangent to \(S^5\) that yields eight-supertwistor formulation of the superparticle's Lagrangian. We find incidence relations of the supertwistors with the \(AdS_5 \times S^5\) superspace coordinates and the set of the quadratic constraints that supertwistors satisfy. It is shown how using the constraints for the (Lorentz-)harmonic variables it is possible to reduce eight-supertwistor formulation to the four-supertwistor one. Respective supertwistors agree with those introduced previously in other models. Advantage of the four-supertwistor formulation is the presence only of the first-class constraints that facilitates analysis of the superparticle model.\(D_k\) gravitational instantons as superpositions of Atiyah-Hitchin and Taub-NUT geometrieshttps://zbmath.org/1472.830252021-11-25T18:46:10.358925Z"Schroers, B. J."https://zbmath.org/authors/?q=ai:schroers.bernd-j"Singer, M. A."https://zbmath.org/authors/?q=ai:singer.michael-aSummary: We obtain \(D_k\) ALF gravitational instantons by a gluing construction which captures, in a precise and explicit fashion, their interpretation as nonlinear superpositions of the moduli space of centred \(SU(2)\) monopoles, equipped with the Atiyah-Hitchin metric, and \(k\) copies of the Taub-NUT manifold. The construction proceeds from a finite set of points in euclidean space, reflection symmetric about the origin, and depends on an adiabatic parameter which is incorporated into the geometry as a fifth dimension. Using a formulation in terms of hyperKähler triples on manifolds with boundaries, we show that the constituent Atiyah-Hitchin and Taub-NUT geometries arise as boundary components of the five-dimensional geometry as the adiabatic parameter is taken to zero.Holographic correlators with multi-particle stateshttps://zbmath.org/1472.830792021-11-25T18:46:10.358925Z"Čeplak, Nejc"https://zbmath.org/authors/?q=ai:ceplak.nejc"Giusto, Stefano"https://zbmath.org/authors/?q=ai:giusto.stefano"Hughes, Marcel R. R."https://zbmath.org/authors/?q=ai:hughes.marcel-r-r"Russo, Rodolfo"https://zbmath.org/authors/?q=ai:russo.rodolfoSummary: We derive the connected tree-level part of 4-point holographic correlators in \(\mathrm{AdS}_3 \times S^3 \times \mathcal{M} \) (where \(\mathcal{M}\) is \(T^4\) or \(K3\)) involving two multi-trace and two single-trace operators. These connected correlators are obtained by studying a heavy-heavy-light-light correlation function in the formal limit where the heavy operators become light. These results provide a window into higher-point holographic correlators of single-particle operators. We find that the correlators involving multi-trace operators are compactly written in terms of Bloch-Wigner-Ramakrishnan functions --- particular linear combinations of higher-order polylogarithm functions. Several consistency checks of the derived expressions are performed in various OPE channels. We also extract the anomalous dimensions and 3-point couplings of the non-BPS double-trace operators of lowest twist at order \(1/c \) and find some positive anomalous dimensions at spin zero and two in the K3 case.Genus zero Gopakumar-Vafa invariants from open stringshttps://zbmath.org/1472.830882021-11-25T18:46:10.358925Z"Collinucci, Andrés"https://zbmath.org/authors/?q=ai:collinucci.andres"Sangiovanni, Andrea"https://zbmath.org/authors/?q=ai:sangiovanni.andrea"Valandro, Roberto"https://zbmath.org/authors/?q=ai:valandro.robertoSummary: We propose a new way to compute the genus zero Gopakumar-Vafa invariants for two families of non-toric non-compact Calabi-Yau threefolds that admit simple flops: Reid's Pagodas, and Laufer's examples. We exploit the duality between M-theory on these threefolds, and IIA string theory with D6-branes and O6-planes. From this perspective, the GV invariants are detected as five-dimensional open string zero modes. We propose a definition for genus zero GV invariants for threefolds that do not admit small crepant resolutions. We find that in most cases, non-geometric T-brane data is required in order to fully specify the invariants.Wrapped brane solutions in Romans \(F(4)\) gauged supergravityhttps://zbmath.org/1472.831052021-11-25T18:46:10.358925Z"Kim, Nakwoo"https://zbmath.org/authors/?q=ai:kim.nakwoo"Shim, Myungbo"https://zbmath.org/authors/?q=ai:shim.myungboSummary: We explore the spectrum of lower-dimensional anti-de Sitter (AdS) solutions in \(F(4)\) gauged supergravity in six dimensions. The ansatz employed corresponds to D4-branes partially wrapped on various supersymmetric cycles in special holonomy manifolds. Re-visiting and extending previous results, we study the cases of two, three, and four-dimensional supersymmetric cycles within Calabi-Yau threefold, fourfold, \(G_2\), and Spin(7) holonomy manifolds. We also report on non-supersymmetric AdS vacua, and check their stability in the consistently truncated lower-dimensional effective action, using the Breitenlohner-Freedman bound. We also analyze the IR behavior and discuss the admissibility of singular flows.Symmetry adapted Gram spectrahedrahttps://zbmath.org/1472.900832021-11-25T18:46:10.358925Z"Heaton, Alexander"https://zbmath.org/authors/?q=ai:heaton.alexander"Hoşten, Serkan"https://zbmath.org/authors/?q=ai:hosten.serkan"Shankar, Isabelle"https://zbmath.org/authors/?q=ai:shankar.isabelleEfficient message transmission via twisted Edwards curveshttps://zbmath.org/1472.940552021-11-25T18:46:10.358925Z"Kırlar, Barış Bülent"https://zbmath.org/authors/?q=ai:kirlar.baris-bulentSummary: In this paper, we suggest a novel public key scheme by incorporating the twisted Edwards model of elliptic curves. The security of the proposed encryption scheme depends on the hardness of solving elliptic curve version of discrete logarithm problem and Diffie-Hellman problem. It then ensures secure message transmission by having the property of one-wayness, indistinguishability under chosen-plaintext attack (IND-CPA) and indistinguishability under chosen-ciphertext attack (IND-CCA). Moreover, we introduce a variant of Nyberg-Rueppel digital signature algorithm with message recovery using the proposed encryption scheme and give some countermeasures to resist some wellknown forgery attacks.Cryptanalysis of a code-based full-time signaturehttps://zbmath.org/1472.940672021-11-25T18:46:10.358925Z"Aragon, Nicolas"https://zbmath.org/authors/?q=ai:aragon.nicolas"Baldi, Marco"https://zbmath.org/authors/?q=ai:baldi.marco"Deneuville, Jean-Christophe"https://zbmath.org/authors/?q=ai:deneuville.jean-christophe"Khathuria, Karan"https://zbmath.org/authors/?q=ai:khathuria.karan"Persichetti, Edoardo"https://zbmath.org/authors/?q=ai:persichetti.edoardo"Santini, Paolo"https://zbmath.org/authors/?q=ai:santini.paolo-mariaSummary: We present an attack against a code-based signature scheme based on the Lyubashevsky protocol that was recently proposed by \textit{Y. Song} et al. [Theor. Comput. Sci. 835, 15--30 (2020; Zbl 1457.94222)]. The private key in the SHMWW scheme contains columns coming in part from an identity matrix and in part from a random matrix. The existence of two types of columns leads to a strong bias in the distribution of set bits in produced signatures. Our attack exploits such a bias to recover the private key from a bunch of collected signatures. We provide a theoretical analysis of the attack along with experimental evaluations, and we show that as few as 10 signatures are enough to be collected for successfully recovering the private key. As for previous attempts of adapting Lyubashevsky's protocol to the case of code-based cryptography, the SHMWW scheme is thus proved unable to provide acceptable security. This confirms that devising secure code-based signature schemes with efficiency comparable to that of other post-quantum solutions (e.g., based on lattices) is still a challenging task.On the existence and construction of maximum distance profile convolutional codeshttps://zbmath.org/1472.940892021-11-25T18:46:10.358925Z"Muñoz Castañeda, Ángel Luis"https://zbmath.org/authors/?q=ai:munoz-castaneda.angel-luis"Plaza-Martín, Francisco J."https://zbmath.org/authors/?q=ai:plaza-martin.francisco-joseSummary: In this paper, we study the conditions for a convolutional code to be MDP in terms of the size of the base field \(\mathbb{F}_q\) as well as the openness of the MDP property in a given family of convolutional codes. Given \((n,k,\delta)\), our main result is an explicit bound depending on \((n,k,\delta)\) such that if \(q\) is greater than this bound, there exists a \((n,k,\delta)\) MDP convolutional code. A similar result is also offered for complete MDP convolutional codes. We show that these bounds are much lower than that those appeared so far in the literature. Finally, we show an explicit and simple construction procedure for MDP convolutional Goppa codes of dimension one.