DOCSLIB.ORG

On Derived Equivalences of K3 Surfaces in Positive Characteristic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Derived Equivalences of K3 Surfaces in Positive Characteristic On Derived Equivalences of K3 Surfaces in Positive Characteristic Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr.rer.nat.) am Fachbereich Mathematik und Informatik der Freien Universit¨atBerlin von Tanya Kaushal Srivastava Berlin, 2018 Betreuerin: Prof. Dr. Dr. h. c. mult. H´el`eneEsnault Erstgutachterin: Prof. Dr. Dr. h. c. mult. H´el`eneEsnault Zweitgutachter: Prof. Dr. Vasudevan Srinivas Tag der Disputation: 21 September 2018 Contents Abstract 1 Zussamengfassung 3 Acknowledgement 5 Chapter 1. Introduction 7 1. Conventions and Notations 8 Chapter 2. Preliminaries on K3 Surfaces 9 1. Basic Properties and the Torelli Theorems 9 2. Moduli Space of Sheaves 15 3. Lifting K3 Surfaces to Characteristic Zero 24 4. Non-Liftable Automorphisms 24 Chapter 3. Derived Equivalences of K3 Surfaces 27 1. Derived Equivalences of Complex K3 Surfaces 28 2. Derived Autoequivalence Group of Complex K3 Surfaces 30 3. Derived Equivalences of K3 Surfaces in Positive Characteristic 32 Chapter 4. Derived Autoequivalences of K3 Surfaces in Positive Characteristic 35 1. Obstruction to Lifting Derived Autoequivalences 35 2. The Cone Inversion Map 44 Chapter 5. Counting Fourier-Mukai Partners in Positive Characteristic 47 1. Fourier-Mukai Partners 47 2. Fourier-Mukai Partners of Ordinary K3 Surfaces 49 3. The Class Number Formula 51 Chapter 6. Appendix: F-crystal on Crystalline Cohomology 53 1. F-(iso)crystals 55 2. The Mukai F-crystal 56 Bibliography 59 Selbstst¨andigkeitserkl¨arung 63 i Abstract For an ordinary K3 surface over an algebraically closed field of positive charac- teristic we show that every automorphism of an ordinary K3 surface lifts to char- acteristic zero. Moreover, we show that the Fourier-Mukai partners of an ordinary K3 surface are in one-to-one correspondence with the Fourier-Mukai partners of the geometric generic fiber of its canonical lift. We also prove that the explicit counting formula for Fourier-Mukai partners of the K3 surfaces with Picard rank two and with discriminant equal to minus of a prime number, in terms of the class number of the prime, holds over a field of positive characteristic as well. We show that the image of the derived autoequivalence group of a K3 surface of finite height in the group of isometries of its crystalline cohomology has index at least two. Moreover, we provide an upper bound on the kernel of this natural cohomological descent map. Further, we give an extended remark in the appendix on the possibility of an F- crystal structure on the crystalline cohomology of a K3 surface over an algebraically closed field of positive characteristic and show that the naive F-crystal structure fails in being compatible with inner product. 1 Zussamengfassung F¨ureine gew¨ohnliche K3-Fl¨ache ¨uber einem algebraisch abgeschlossenen K¨orper positiver Charakteristik zeigen wir, dass jeder Automorphismus einer gew¨ohnlichen K3-Fl¨ache zu Charakteristik Null liftet. Dar¨uber hinaus zeigen wir, dass die Fourier- Mukai-Partner einer gew¨ohnlichen K3-Fl¨ache in einer eins-zu-eins Korrespondenz mit den Fourier-Mukai-Partnern der geometrischen generischen Faser ihres kanonis- chen lift sind. Wir beweisen auch, dass die explizite Z¨ahlformelf¨urFourier-Mukai- Partner der K3-Fl¨achen mit Picard-Rang zwei und mit Diskriminante gleich mi- nus einer Primzahl, bezogen auf die Klassenzahl der Primzahl, ¨uber einem K¨orper positiver Charakteristik g¨ultig ist. Allgemeiner zeigen wir, dass das Bild der deriv- ierten auto¨aquivalenzgruppe einer K3-Fl¨ache, mit endlicher H¨ohe,in der Gruppe der Isometrien auf ihrer kristallinen Kohomologie mindestens von Index zwei ist. Außerdem geben wir eine Obergrenze f¨urden Kern der nat¨urlichen Abbildung von der derivierten Auto¨aquivalenzgruppe einer K3-Fl¨ache zu der Gruppe der Isome- trien der kristallinen Kohomologie. Des Weiteren machen wir im Anhang eine ausf¨uhrliche Bemerkung ber die M¨oglichkeit einer F-Kristallstruktur auf der kristallinen Kohomologie einer K3- Fl¨ache ber einem algebraisch abgeschlossenen K¨orper positiver Charackeristik und zeigen, dass die naive F-Kristallstruktur nicht mit dem Innenprodukt kompatibel ist. 3 Acknowledgement I owe a special gratitude to my supervisor Prof. Dr. H´el`eneEsnault, for her guidance and support. I am grateful to her for introducing me to research and for the continuous encouragement and inspiration she always provided me. I thank Berlin Mathematical School for the PhD fellowship. I would like to say a big thanks to Dr. Michael Groechenig who was always there for a mathematical discussion, even at short notices and to hear my endless excitement about mathematics and complains in general. I am greatly thankful to Prof. Dr. Fran¸coisCharles, Prof. Dr. Christian Liedtke, Prof. Dr. Daniel Huybrechts, Prof. Dr. Martin Olsson, Prof. Dr. Max Lieblich, Prof. Dr. Lenny Taelman, Prof. Dr. Sofia Tirabassi and Prof. Dr. Gabriele Vessozi for many mathematical discussions and their patience in answering my questions and suggestions. I am really thankful to Prof. Dr. Vasudevan Srinivas for his corrections, suggestions and comments on the thesis. I would like to thank Dr. Kay R¨ulling,Dr. Fabio Tonini, Dr. Raju Krish- namoorthy, Dr. Lei Zhang, Dr. Simon Pepin Lehalleur, Dr. Marta Pieropan and Dr. Enlin Yang from whom I learnt a lot of mathematics. A special mention of my office mates and friends Marcin Lara, Pedro Angel Castillejo Blasco, Marco d'Addezio, Yun Hao, Efstathia Katsigianni, Dr. Giulia Battiston, Dr. Elena La- vanda, Wouter Zomervrucht and Dr. Maciek Zdanowicz for their supply of hugs, motivation and mathematics. Lastly, I am grateful to my parents who were always there for me. 5 CHAPTER 1 Introduction The derived category of coherent sheaves on a smooth projective variety was first studied as a geometrical invariant by Mukai in the early 1980's. In case the smooth projective variety has an ample canonical or anti-canonical bundle, Bondal- Orlov [13] proved that, if two such varieties have equivalent bounded derived cat- egories of coherent sheaves, then they are isomorphic. However, in general this is not true. The bounded derived category of coherent sheaves is not an isomor- phism invariant. Mukai [54] showed that for an Abelian variety over C, its dual has equivalent bounded derived category. Moreover, in many cases it can be shown that the dual of an Abelian variety is not birational to it, which implies that derived categories are not even birational invariants, see [31] Chapter 9. Similarly, Mukai showed in [55] that for K3 surfaces over C, there are non-isomorphic K3 surfaces with equivalent derived categories. This led to the natural question of classifying all derived equivalent varieties. For K3 surfaces, the case of interest to us, this was completed over C in late 1990's by Mukai and Orlov (see Theorem 3.8) using Hodge theory along with the Global Torelli Theorem (see Theorem 2.5). As a consequence, it was shown that there are only finitely many non-isomorphic K3 surfaces with equivalent bounded derived categories (see Proposition 3.10) and a counting formula was also proved (see Theorem 3.12). On the other hand, for K3 surfaces over a field of positive characteristic, a partial answer to the classification question was first given by Lieblich-Olsson [46] (see Theorem 3.20) in early 2010's. They showed that there are only finitely many non-isomorphic K3 surfaces with equivalent bounded derived categories. We remark here that due to unavailability of a positive characteristic version of the global Torelli Theorem for K3 surfaces of finite height, it is currently not feasible to give a complete cohomological description of derived equivalent K3 surfaces. However, a description in terms of moduli spaces was given by Lieblich- Olsson. We also point out here that the proofs of these results go via lifting to characteristic zero and thus use the Hodge theoretic description given by Mukai and Orlov. Furthermore, Lieblich-Olsson [47] also proved the derived version of the Torelli theorem (see Theorem 3.23) using the Crystalline Torelli theorem for supersingular K3 surfaces (see Theorem 2.13). Meanwhile in 1990's another school of thought inspired by string theory in physics led Kontsevich [42] to propose the homological mirror symmetry conjec- ture which states that the bounded derived category Db(X) of coherent sheaves of a projective variety X is equivalent (as a triangulated category) to the bounded derived category DbF uk(X;ˇ β) of the Fukaya category F uk(X;ˇ β) of a mirror Xˇ with its symplectic structure β. Moreover, the symplectic automorphisms of Xˇ induce derived autoequivalences of Db(X). This provided a natural motivation for the study of the derived autoequivalence group. 7 8 1. INTRODUCTION For K3 surfaces X over C, the structure of the group of derived autoequivalences was analyzed by Ploog in [65], Hosono et al. in [29] and Huybrechts, et al. in [33]. They showed that the image of Aut(Db(X)) under the homomorphism b Aut(D (X)) ! OHodge(H^(X; Z)); where OHodge(H^(X; Z)) is the group of Hodge isometries of the Mukai lattice of X, has index 2. However, the kernel of this map has a description only in the special case when the Picard rank of X is 1, given by [6] (also see Section 2). In this thesis, we study the above two questions in more details for the case of K3 surfaces over an algebraically closed field of positive characteristic. We collect most of the preliminary material about K3 surfaces in Chapter 2 and about derived equivalences in Chapter 3. In Chapter 4, we address the question on the group of derived autoequivalences for K3 surfaces of finite height. We show that the image of the derived autoequivalence group of a K3 surface of finite height in the group of isometries of its crystalline cohomology has index at least two (Theorem 4.29).
Load more

Recommended publications
  • Automorphisms of Minimal Entropy on Supersingular K3 Surfaces
    AUTOMORPHISMS OF MINIMAL ENTROPY ON SUPERSINGULAR K3 SURFACES SIMON BRANDHORST, VÍCTOR GONZÁLEZ-ALONSO Abstract. In this article we give a strategy to decide whether the logarithm of a given Salem number is realized as entropy of an automorphism of a su- persingular K3 surface in positive characteristic. As test case it is proved that log λd, where λd is the minimal Salem number of degree d, is realized in char- acteristic 5 if and only if d ≤ 22 is even and d 6= 18. In the complex projective setting we settle the case of entropy log λ12, left open by McMullen, by giv- ing the construction. A necessary and sufficient test is developed to decide whether a given isometry of a hyperbolic lattice, with spectral radius bigger than one, is positive, i.e. preserves a chamber of the positive cone. Contents 1. Introduction 1 Notation 4 Acknowledgements 4 2. Lattices 4 3. Torelli theorems for supersingular K3 surfaces 5 4. Positivity 8 5. Lattices in number fields 11 6. Gluing lattices and isometries 12 7. Realizability Conditions 14 8. Realized Salem numbers 18 Appendix A. Realizing λ12 over C 25 References 25 arXiv:1609.02716v2 [math.AG] 20 Jul 2017 1. Introduction A Salem number is a real algebraic integer λ > 1 which is conjugate to 1/λ and whose other conjugates lie on the unit circle. In each even degree d there is a unique smallest Salem number λd. Conjecturally the smallest Salem number is λ10, found by Lehmer in 1933 [17]. Date: September, 2016. 2010 Mathematics Subject Classification.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Moduli Spaces And
    Mathematisches Forschungsinstitut Oberwolfach Report No. 23/2016 DOI: 10.4171/OWR/2016/23 Moduli spaces and Modular forms Organised by Jan Hendrik Bruinier, Darmstadt Gerard van der Geer, Amsterdam Valery Gritsenko, Villeneuve d’Ascq 24 April – 30 April 2016 Abstract. The roots of both moduli spaces and modular forms go back to the theory of elliptic curves in the 19th century. Both topics have seen an enormous growth in the second half of the 20th century, but the interaction between the two remained limited. Recently there have been new develop- ments that led to new points of contact between the two topics. One is the theory of K3 surfaces that is rapidly gaining a lot of new interest. Here the link with modular forms on orthogonal groups has led to progress on the Kodaira dimension of the moduli spaces of K3 surfaces. Another new devel- opment has been the use of moduli spaces of curves to gather new information about Siegel modular forms. The workshop intended to bring representatives from both the theory of moduli and the theory of modular forms together to further the interaction between the two topics as the time seemed ripe to do this. Mathematics Subject Classification (2010): 11xx, 14xx. Introduction by the Organisers The workshop Moduli Spaces and Modular Forms, organized by Jan Bruinier (Darmstadt), Gerard van der Geer (Amsterdam) and Val´ery Gritsenko (Lille) was held 25-29 April, 2016 and was attended by 52 participants from all over the world. The attendance ranged from senior leaders in the field to young postdocs and advanced Ph.D.
    [Show full text]
  • Supersingular K3 Crystals Astérisque, Tome 64 (1979), P
    Astérisque ARTHUR OGUS Supersingular K3 crystals Astérisque, tome 64 (1979), p. 3-86 <http://www.numdam.org/item?id=AST_1979__64__3_0> © Société mathématique de France, 1979, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ SUPERSINGULAR K3 CRYSTALS by Arthur OGUS (Berkeley) §0. INTRODUCTION. This paper is intented as propaganda for the machinery of crystalline cohomolo- gy, and in particular for the philosophy that F-crystals are a partial analogue, in characteristic p, to Hodge structures in characteristic zero. An extremely rudimen­ tary start along this road, for "abstract" F-crystals and Hodge structures, was made in [15] ; here we turn to crystals arising geometrically, especially from supersingu- lar abelian varieties and K3 surfaces. As we shall see, it is reasonable to hope that the moduli of such varieties are given by the moduli of their F-crystals, which in fact form explicit "period-spaces". Here is a plan of the paper : The first section contains some refinements of gene­ rally known facts concerning crystalline Chern classes, e.g. an integral version of Bloch's theorem relating flat and crystalline cohomology (1.7), Conditions guaran- teeing that c : Pic <8> 2£/p2£ H,™ is injective (1.4), and a formula for certain second order obstructions to extending invertible sheaves in a family (l.
    [Show full text]
  • Supersingular K3 Surfaces for Large Primes
    Supersingular K3 surfaces for large primes The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Maulik, Davesh. “Supersingular K3 Surfaces for Large Primes.” Duke Mathematical Journal, 163, 13 (October 2014): 2357–2425 As Published http://dx.doi.org/10.1215/00127094-2804783 Publisher Duke University Press Version Original manuscript Citable link http://hdl.handle.net/1721.1/115923 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ SUPERSINGULAR K3 SURFACES FOR LARGE PRIMES DAVESH MAULIK WITH AN APPENDIX BY ANDREW SNOWDEN ABSTRACT. Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height) then its Picard rank is 22. Along with work of Nygaard- Ogus, this conjecture implies the Tate conjecture for K3 surfaces over finite fields with p ≥ 5. We prove Artin’s conjecture under the additional assumption that X has a polarization of degree 2d with p> 2d + 4. Assuming semistable reduction for surfaces in characteristic p, we can improve the main result to K3 surfaces which admit a polarization of degree prime-to-p when p ≥ 5. The argument uses Borcherds’ construction of automorphic forms on O(2, n) to construct am- ple divisors on the moduli space. We also establish finite-characteristic versions of the positivity of the Hodge bundle and the Kulikov-Pinkham-Persson classification of K3 degenerations. In the appendix by A. Snowden, a compatibility statement is proven between Clifford constructions and integral p-adic comparison functors.
    [Show full text]
  • Arxiv:1908.07928V3 [Math.AG] 15 Dec 2020 Characteristic
    ON AUTOMORPHISMS AND THE CONE CONJECTURE FOR ENRIQUES SURFACES IN ODD CHARACTERISTIC LONG WANG Abstract. We prove that, for an Enriques surface in odd character- istic, the automorphism group is finitely generated and it acts on the effective nef cone with a rational polyhedral fundamental domain. We also construct a smooth projective surface in odd characteristic which is birational to an Enriques surface and whose automorphism group is discrete but not finitely generated. 1. Introduction Morrison [Mo93] and Kawamata [Ka97] proposed the following conjecture concerning the structure of the effective nef cone of a Calabi{Yau variety (i.e., a normal projective variety with mild singularities whose canonical divisor is numerically trivial). Conjecture 1.1 (Cone conjecture). Let X be a Calabi{Yau variety over an algebraically closed field. Then there exists a rational polyhedral cone Π which is a fundamental domain for the action of the automorphism group Aut(X) on the effective nef cone Nefe(X) := Nef(X) \ Eff(X), in the sense e S ∗ ◦ ∗ ◦ ∗ that Nef (X) = g2Aut(X) g Π, and Π \ (g Π) = ? unless g = id. Over the field of complex numbers, the conjecture was verified in the 2- dimensional case (see [To10]). However, even for smooth Calabi{Yau three- folds, the conjecture is widely open. We refer to [LOP18] for a survey. In this paper, we concentrate on the case of Enriques surfaces over a field of odd characteristic. The restriction on the characteristic is used to ensure that these surfaces admit an ´etalecover by a K3 surface. Recently Lieblich{Maulik [LM18] proved the cone conjecture for K3 surfaces in odd arXiv:1908.07928v3 [math.AG] 15 Dec 2020 characteristic.
    [Show full text]
  • [Math.AG] 1 Dec 2005 Obepit Ftype of Points Double Nouin H Utetsurface Quotient the Involution
    KUMMER SURFACES FOR THE SELFPRODUCT OF THE CUSPIDAL RATIONAL CURVE STEFAN SCHROER¨ Revised version, 11 November 2005 Abstract. The classical Kummer construction attaches to an abelian surface a K3 surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersin- gular abelian surfaces by the selfproduct of the rational cuspidal curve, and the sign involution by suitable infinitesimal group scheme actions, I give the correct Kummer-type construction for this situation. We encounter rational double points of type D4 and D8, instead of type A1. It turns out that the resulting surfaces are supersingular K3 surfaces with Artin invariant one and two. They lie in a 1-dimensional family obtained by simultaneous resolution, which exists after purely inseparable base change. Contents Introduction 1 1. Generalities on αp-actions 3 2. Quotients, singularities, and base change 5 3. The cuspidal rational curve 9 4. The selfproduct for the cuspidal rational curve 11 5. The singularity coming from the quadruple point 13 6. Singularities coming from fixed points 15 7. Quasielliptic fibrations 17 8. K3 surfaces and rational surfaces 20 9. Discriminants and Artin invariants 23 10. Blowing up curves on rational singularities 24 11. Blowing ups in genus-one fibrations 27 arXiv:math/0504023v3 [math.AG] 1 Dec 2005 12. Simultaneous resolutions and nonseparatedness 29 13. Isomorphic fibers in the family 31 References 32 Introduction Let A be an abelian surface over the complex numbers, and ι : A A the sign involution. The quotient surface Z = A/ι is a normal surface with→ 16 rational double points of type A1, whose minimal resolution X is a K3 surface.
    [Show full text]
  • The Tate Conjecture for K3 Surfaces Over Finite Fields
    Invent math DOI 10.1007/s00222-012-0443-y The Tate conjecture for K3 surfaces over finite fields François Charles Received: 6 July 2012 / Accepted: 29 November 2012 © Springer-Verlag Berlin Heidelberg 2012 Abstract Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p ≥ 5. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p ≥ 5. Our results also yield the Tate conjec- ture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p ≥ 5. Mathematics Subject Classification (2010) 14C22 · 14C25 · 14G15 · 14J28 1 Introduction The goal of this paper is to study the Tate conjecture for varieties with h2,0 = 1 over finite fields. The main result is the following. Recall that Artin conjectured in [3] that the rank of the Néron-Severi group of a supersingular K3 surface over a finite field—in the sense of Artin, that is, a K3 surface whose formal Brauer group has infinite height—has the maximal possible value, that is, 22. Theorem 1 Artin’s conjecture holds for supersingular K3 surfaces over al- gebraically closed fields of characteristic p ≥ 5. F. Charles () IRMAR–UMR 6625 du CNRS, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France e-mail: [email protected] F. Charles Let X be a smooth projective variety over a finite field k.Let be a prime number different from the characteristic of k.
    [Show full text]
  • Moduli of Supersingular Enriques Surfaces
    TECHNISCHE UNIVERSITAT¨ MUNCHEN¨ Fakultat¨ fur¨ Mathematik Lehr- und Forschungseinheit Algebra — Algebraische Geometrie Moduli of supersingular Enriques surfaces Kai Behrens Vollstandiger¨ Abdruck der von der Fakultat¨ fur¨ Mathematik der Technischen Uni- versitat¨ Munchen¨ zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Prof. Dr. Ulrich Bauer Prufer¨ der Dissertation: 1. Prof. Dr. Christian Liedtke 2. Prof. Dr. Andreas Rosenschon Ludwig-Maximilians Universitat¨ Munchen¨ Die Dissertation wurde am 20.04.2020 bei der Technischen Universitat¨ Munchen¨ eingereicht und durch die Fakultat¨ fur¨ Mathematik am 02.07.2020 angenommen. Moduli of supersingular Enriques surfaces Kai Behrens Contents Introduction . 5 Motivation......................................................................... 5 Results............................................................................ 7 Acknowledgements . 13 Chapter 0. Basic facts on Grassmannians, lattices and K3 surfaces . 15 x 1. Charts of Grassmannian varieties . 15 x 2. Lattices . 16 x 3. K3 surfaces and Enriques surfaces . 16 x 4. Supersingular K3 surfaces . 18 x 5. Ogus’ Crystalline Torelli theorem for supersingular K3 surfaces . 19 Chapter 1. On the number of Enriques quotients of a K3 surface . 25 x 1. Enriques quotients of K3 surfaces of finite height . 25 x 2. The supersingular case . 27 x 3. Existence of Enriques quotients for supersingular K3 surfaces . 34 Chapter 2. A fibration on the period space of supersingular K3 surfaces . 53 x 1. Background . 53 x 2. Existence of a rational map Mσ 99K Mσ−1 ..................................... 54 x 3. Extending the rational map to a morphism Mσ !Mσ−1 ......................... 58 x 4. The proof of Lemma 2.8 . 60 x 5. The fibers of the morphisms Φi ................................................. 71 Chapter 3. A moduli space for supersingular Enriques surfaces .
    [Show full text]
  • Twisted Derived Equivalences and Isogenies Between K3 Surfaces In
    TWISTED DERIVED EQUIVALENCES AND ISOGENIES BETWEEN K3 SURFACES IN POSITIVE CHARACTERISTIC DANIEL BRAGG AND ZIQUAN YANG Abstract. We study isogenies between K3 surfaces in positive characteristic. Our main result is a char- acterization of K3 surfaces isogenous to a given K3 surface X in terms of certain integral sublattices of the second rational ℓ-adic and crystalline cohomology groups of X. This is a positive characteristic analog of a result of Huybrechts [15], and extends results of [52]. We give applications to the reduction types of K3 surfaces and to the surjectivity of the period morphism. To prove these results we describe a theory of B-fields and Mukai lattices in positive characteristic, which may be of independent interest. We also prove some results on lifting twisted Fourier–Mukai equivalences to characteristic 0, generalizing results of Lieblich and Olsson [29]. Contents 1. Introduction 1 2. B–fields and the twisted Mukai lattice in positive characteristic 5 3. Twisted Chern characters and action on cohomology 15 4. Rational Chow motives and isogenies 17 5. Lifting derived isogenies to characteristic 0 18 6. Existence Theorems 21 7. Uniqueness Theorems 29 8. Isogenies and Hecke Orbits 33 References 37 1. Introduction ∼ An isogeny between two K3 surfaces X and X′ is an isomorphism h2(X′) −→ h2(X) of rational Chow motives which preserves the cohomological pairing (see Def. 4.1). An isogeny has cohomological realizations, 2 ′ ∼ 2 2 which are isomorphisms H (X )Q −→ H (X)Q, where H (−)Q denotes the second rational Betti, ℓ–adic, or crystalline cohomology group. A natural Torelli–type problem is to understand which isomorphisms of arXiv:2102.01193v1 [math.AG] 1 Feb 2021 cohomology groups are induced by isogenies.
    [Show full text]
  • Complex Algebraic Geometry
    Mathematisches Forschungsinstitut Oberwolfach Report No. 47/2011 DOI: 10.4171/OWR/2011/47 Complex Algebraic Geometry Organised by Fabrizio Catanese, Bayreuth Yujiro Kawamata, Tokyo Bernd Siebert, Hamburg Gang Tian, Princeton September 25th – October 1st, 2011 Abstract. The conference focused on several classical and modern topics in the realm of complex algebraic geometry, such as moduli spaces, bira- tional geometry and the minimal model program, Mirror symmetry and Gromov-Witten invariants, Hodge theory, curvature flows, algebraic surfaces and curves. Mathematics Subject Classification (2000): 14xx , 18xx , 32xx , 53xx . Introduction by the Organisers The workshop Complex Algebraic Geometry, organized by Fabrizio Catanese (Bay- reuth), Yujiro Kawamata (Tokyo), Bernd Siebert (Hamburg) and Gang Tian (Princeton), drew together 56 participants from all continents (...). There were several young PhD students and other PostDocs in their 20s and early 30s, to- gether with established leaders of the fields related to the thematic title of the workshop. There were 23 50-minutes talks, each followed by a lively discussion. As usual at an Oberwolfach Meeting, the mathematical discussions continued outside the lecture room throughout the day and the night. The Conference fully realized its purported aim, of setting in contact mathematicians with different specializations and non uniform background, of presenting new fashionable topics alongside with new insights on long standing classical open problems, and also cross-fertilizations with other research topics. Many talks were devoted to classification theory, especially to the birational geometry of 3-folds, but also to surfaces and higher dimensional varieties, Fano manifolds, relations with mirror symmetry. 2670 Oberwolfach Report 47/2011 Another important direction was moduli spaces: their rationality properties, moduli spaces of curves and Gromov-Witten invariants, automorphisms.
    [Show full text]
  • Arxiv:1808.05851V5 [Math.AG] 27 Nov 2020 S Characteristic 1.1
    SUPERSINGULAR IRREDUCIBLE SYMPLECTIC VARIETIES LIE FU AND ZHIYUAN LI Abstract. In complex geometry, irreducible holomorphic symplectic varieties, also known as com- pact hyper-K¨ahler varieties, are natural higher-dimensional generalizations of K3 surfaces. We pro- pose to study such varieties defined over fields of positive characteristic, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of characteristic p> 0, under natural numerical conditions: (1) smooth moduli spaces of semistable (twisted) sheaves on K3 surfaces, (2) smooth Albanese fibers of moduli spaces of semistable sheaves on abelian surfaces. Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to be equivalent in both cases (1) and (2). Their equivalence is expected in general. On the geometric aspect, we conjecture that unirationality characterizes supersingularity for symplectic varieties. Such an equivalence is established in case (1), assuming the same is true for K3 surfaces. In case (2), we show that rational chain connectedness is equivalent to supersingularity. On the motivic aspect, we conjecture that the algebraic cycles on supersingular symplectic varieties should be much simpler than their complex counterparts: its rational Chow motive is of supersingular abelian type, the rational Chow ring is representable and satisfies the Bloch– Beilinson conjecture and Beauville’s splitting property. As evidences, we prove all these conjectures on algebraic cycles for supersingular varieties in both cases (1) and (2). 1. Introduction This paper is an attempt to generalize to higher dimensions the beautiful theory of supersingular K3 surfaces.
    [Show full text]
  • Isogenies Between K3 Surfaces Over ¯ Fp
    ¯ Isogenies between K3 Surfaces over Fp Ziquan Yang Abstract We generalize Mukai and Shafarevich’s definitions of isogenies between K3 surfaces over C to an arbitrary perfect field and describe how to construct isogenous K3 surfaces ¯ over Fp by prescribing linear algebraic data when p is large. The main step is to show that isogenies between Kuga-Satake abelian varieties induce isogenies between K3 surfaces, in the context of integral models of Shimura varieties. As a byproduct, we show that every K3 surface of finite height admits a CM lifting under a mild assumption on p. 1 Introduction Efforts to define the notion of isogeny between K3 surfaces have a long history. We refer the reader to [33] for a summary. Shafarevich defined an isogeny between two complex algebraic ∼ K3 surfaces X; X0 to be a Hodge isometry H2(X0; Q) ! H2(X; Q) (cf. [42]). Mukai reserved the term for those Hodge isometries that are induced by correspondences (cf. [34]). Thanks to a recent result of Buskin [10, Theorem 1.1], now we know these two definitions coincide. Therefore, we make the following definition, which generalizes Mukai and Shafarevich’s: Definition 1.1. Let k be a perfect field with algebraic closure k¯. Let X; X0 be two K3 surfaces over k with a correspondence (i.e., algebraic cycles on X × X0 over k with Q- 0 coefficients) f : X X . 2 0 ∼ • If char k = 0, we say f is an isogeny over k if the induced map Hét(Xk¯; Af ) ! 2 Hét(Xk¯; Af ) is an isometry.
    [Show full text]