Numerically Trivial Dualizing Sheaves Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨atD¨usseldorf vorgelegt von Andr´eSchell aus Duisburg D¨usseldorf, Dezember 2019 aus dem Mathematischen Institut der Heinrich-Heine-Universit¨atD¨usseldorf Gedruckt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨atder Heinrich-Heine-Universit¨atD¨usseldorf Referent: Prof. Dr. Stefan Schr¨oer Korreferent: Juniorprof. Dr. Marcus Zibrowius Tag der m¨undlichenPr¨ufung:30.01.2020 Summary Motivation for this work was the question to what extent the Beauville{Bogomolov decom- position, or parts and variants of it, holds true in positive characteristic. The statement itself applies to compact K¨ahlermanifolds and its proof uses differential geometry. It can be transferred to smooth projective schemes X over the complex numbers by the Riemann existence theorem. Under the assumption that the dualizing sheaf !X is numerically triv- ial, it states that X admits a finite ´etalecovering X0 ! X such that the total space 0 X satisfies !X0 'OX0 and decomposes in a special way as a product, uniquely up to permutation. During the first, preparatory part of this thesis, the notion of numerical triviality is treated and characterized. The Albanese morphism serves as a central technical tool in what follows, so its core properties are developed in a self-contained way, also verifying its existence under more general assumptions than what seems to be recorded in literature. The second, main part begins to answer the question whether !X 2 Pic(X) has finite order as a consequence of its numerical triviality. More generally, this question can be asked if !X 'OX (KX ) is not necessarily invertible, as long as the canonical divisor KX is a Weil divisor. The main result is an affirmative answer for Q-Gorenstein surfaces with the property that the pullback of KX along a resolution of singularities continues to be a Weil divisor. The proof adapts ideas used by Sakai to extend the Enriques classification of smooth surfaces to normal Gorenstein surfaces over the complex numbers. Without the Q-Gorenstein assumption, the conclusion fails to hold true. Counterexamples of surfaces are given such that KX is numerically trivial of infinite order. Once it is known that !X 2 Pic(X) has finite order d, there always exists a finite flat 0 covering X ! X such that !X0 'OX0 . It is ´etaleif the characteristic p of the ground field is not dividing d. Here the main result is that for normal X, such an ´etalecovering conversely cannot exist if the latter condition is violated. If the value of d is known, this especially gives a criterion for excluding the existence of a decomposition on an ´etale covering of X. For smooth surfaces, the situation is analyzed in detail afterwards. A softened variant of the decomposition, holding on a finite flat covering, is suggested. Such a decomposition is in turn proved to be unique, using a result by Fujita, and it now exists for all smooth surfaces in arbitrary characteristic. i Zusammenfassung Motivation f¨urdiese Arbeit war die Frage, inwiefern die Beauville-Bogomolov-Zerlegung, oder Teile und Varianten von ihr, weiterhin in positiver Charakteristik g¨ultigist. Die Aussage selbst gilt f¨urkompakte K¨ahler-Mannigfaltigkeiten und ihr Beweis benutzt Dif- ferentialgeometrie. Sie kann auf glatte projektive Schemata X ¨uber den komplexen Zahlen mithilfe des Riemannschen Existenzsatzes ¨ubertragen werden. Unter der Annahme, dass die dualisierende Garbe !X numerisch trivial ist, sagt sie die Existenz einer endlichen ¨ 0 0 ´etalen Uberlagerung X ! X mit !X0 'OX0 aus, sodass der Totalraum X eine spezielle Produktzerlegung erlaubt, welche eindeutig bis auf Permutation ist. Im ersten, vorbereitenden Teil dieser Dissertation wird der Begriff der numerischen Trivialit¨atbehandelt und charakterisiert. Der Albanese-Morphismus dient im Weiteren als zentrales technisches Werkzeug, sodass seine Kerneigenschaften in eigenst¨andigerWeise entwickelt werden. Seine Existenz kann dabei auch unter allgemeineren Voraussetzungen, als es anscheinend in der Literatur festgehalten ist, nachgewiesen werden. Der Hauptteil dieser Arbeit beginnt die Frage zu beantworten, ob !X 2 Pic(X) stets endliche Ordnung als Konsequenz ihrer numerischen Trivialit¨atbesitzt. Allgemeiner kann diese Frage gestellt werden, falls !X 'OX (KX ) nicht notwendigerweise invertierbar ist, solange der kanonische Divisor KX ein Weil-Divisor ist. Das Hauptresultat ist eine posi- tive Antwort f¨ur Q-Gorenstein-Fl¨achen mit der Eigenschaft, dass der Pullback von KX entlang einer Aufl¨osungder Singularit¨atenein Weil-Divisor bleibt. Der Beweis verwendet angepasste Ideen von Sakai, welche dieser zur Erweiterung der Enriques-Klassifikation auf normale Gorenstein-Fl¨achen ¨uber den komplexen Zahlen verwendet hat. Ohne die Q- Gorenstein-Eigenschaft bleibt die Schlussfolgerung nicht g¨ultig.Es werden Gegenbeispiele von Fl¨achen mit numerisch trivialem KX von unendlicher Ordnung gegeben. Sobald bekannt ist, dass !X 2 Pic(X) endliche Ordnung d besitzt, existiert eine endliche ¨ 0 flache Uberlagerung X ! X mit der Eigenschaft !X0 'OX0 . Diese ist ´etale,wenn die Charakteristik p des Grundk¨orpers kein Teiler von d ist. Hier ist das Hauptresultat, dass f¨urnormale X eine solche ´etale Uberlagerung¨ umgekehrt nicht existieren kann, falls letztere Bedingung verletzt ist. Ist der Wert d bekannt, bildet dies ein Ausschlusskriterium f¨urdie Existenz der Zerlegung nach einer ´etalen Uberlagerung¨ von X. F¨urglatte Fl¨achen wird die Situation anschließend detailliert analysiert. Es wird eine mildere Variante der Zerlegung vorgeschlagen, welche nach einer endlichen flachen Uberlagerung¨ existiert. Eine derartige Zerlegung wird wiederum als eindeutig bewiesen, unter Verwendung eines Resultates von Fujita, und sie existiert nun f¨uralle glatten Fl¨achen in beliebiger Charakteristik. iii Contents Contents Introduction 1 Part I 9 1 Numerically Trivial Sheaves 11 1.1 Regularity Conditions . 12 1.2 Etale´ Cohomology . 15 1.3 `-adic Cohomology . 16 1.4 Comparison to Other Cohomology Theories . 18 1.5 First Chern Classes . 21 1.6 Intersection Numbers . 25 1.7 Characterization of Numerical Triviality . 30 1.8 Chow Ring . 32 1.9 Higher Chern Classes . 33 1.10 Cycle Map . 39 2 Dualizing Sheaves 41 2.1 Serre Duality . 41 2.2 K¨ahlerDifferentials . 44 2.3 Grothendieck Duality . 45 3 Albanese Morphisms 53 3.1 Poincar´eSheaves . 54 3.2 Abelian Schemes . 55 3.3 Albanese Schemes . 58 3.4 Albanese Torsors . 71 v Contents Part II 83 4 Dualizing Sheaves of Finite Order 85 4.1 Introductory Examples . 86 4.2 Picτ (X) is a Torsion Group . 90 4.3 Maximal Albanese Dimension . 90 4.4 Kodaira Dimension and the Enriques Classification . 93 4.5 Smooth Surfaces with Numerically Trivial Dualizing Sheaf . 99 4.6 Q-Gorenstein Surfaces with Numerically Trivial Canonical Divisor . 107 4.7 Examples and Counterexamples of Singular Surfaces . 117 5 Invertible Sheaves and Coverings 121 5.1 Canonical Coverings Associated to Invertible Sheaves . 121 5.2 Non-Existence of Etale´ Coverings . 126 6 Total Spaces of Coverings 129 6.1 Uniqueness of a Decomposition . 129 6.2 Surfaces and the Beauville{Bogomolov Decomposition . 131 A Appendix 139 A.1 Basic Group Schemes . 139 A.2 Principal Homogeneous Spaces . 141 A.3 Analytification and GAGA . 142 A.4 Cohomology and Base Change . 144 A.5 Fibrations and Zariski's Main Theorem . 145 A.6 Curves of Fiber Type . 150 A.7 Picard Schemes . 152 A.8 Approach Using Models . 156 Bibliography 159 List of Tables and Figures 169 Index 171 vi Introduction Introduction Algebraic geometry originated from the challenge to study the space of common zeros X = V (f1; : : : ; fr) of polynomials f1; : : : ; fr in several indeterminates with coefficients in a field k. Methods from commutative algebra and geometry are applied in synergy to make progress, and also these single branches profit by this interplay. In the 1960s, Grothendieck introduced the notion of a scheme X, which drastically enlarged the framework. The additional datum of a structure sheaf OX on X associates in a compatible way to each open 0 subset U ⊂ X a ring H (U; OX ), reflecting this connection. One central object on proper X is the dualizing sheaf !X , which contains both algebraic and geometric information: Assuming that X is sufficiently regular, then on the algebraic side, it yields a duality i _ n−i _ H (X; E ⊗ !X ) = H (X; E) on the cohomology groups of locally free sheaves E on X. Cohomology groups are substantial invariants of a scheme, for example their disparity on two schemes already implicates that both schemes themselves must be different. Moreover, 1 1 !X = det(ΩX ) is the determinant of the cotangent sheaf ΩX , where the latter includes essential geometric data. The coincidence !X 'OX of the dualizing sheaf and the structure sheaf is a special situation with multilayer consequences for the scheme X. For instance, !X can be omitted from the duality above. In the case that the dualizing sheaf is different from the structure sheaf, it is desirable to measure the discrepancy and also to slightly modify X in order to ⊗d move both sheaves closer together. A weaker variant is to ask if some power !X 'OX of the dualizing sheaf is trivial. Taking another step back, !X can be numerically trivial, meaning that its intersection number with every curve C ⊂ X is zero. Figure 1: Real points around the origin of two smooth surfaces X with !X 'OX . 1 Introduction Over the field k = C of complex numbers, the situation was clarified using methods from differential geometry. Although the techniques available in this different branch of mathematics cannot be directly applied in algebraic geometry, it is possible to transfer the conclusion. First, consider the result itself. The Beauville{Bogomolov decomposition states that every connected, compact K¨ahler an 2 Q 0 manifold Y with c1 (!Y ) = 0 in H (Y ; ) admits a finite ´etalecovering Y ! Y such that 0 Qr 0 the total space Y = i=1 Yi decomposes as a product, where each factor is a complex torus, a Calabi{Yau manifold or a Hyperk¨ahlermanifold.