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Coherent Sheaves on Calabi-Yau Manifolds, Picard-Fuchs Equations and Potential Functions Dissertation Zur Erlangung Des Doktorgr

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Coherent Sheaves on Calabi-Yau Manifolds, Picard-Fuchs Equations and Potential Functions Dissertation Zur Erlangung Des Doktorgr Coherent Sheaves on Calabi-Yau manifolds, Picard-Fuchs equations and potential functions Dissertation zur Erlangung des Doktorgrades der Fakult¨atf¨urMathematik und Physik der Albert-Ludwigs-Universit¨at Freiburg im Breisgau vorgelegt von Kathrin Natalie Peternell Februar 2018 Dekan der Fakult¨atf¨urMathematik und Physik Prof. Dr. Gregor Herten Erstgutachter Priv.-Doz. Dr. Emanuel Scheidegger Zweitgutachter Prof. Sheldon Katz, PhD Datum der Promotionspr¨ufung 21. Juni 2018 Contents 1 Introduction 1 2 Preliminaries 15 2.1 Notation . 15 2.2 Singularities . 15 2.3 Projective fibre spaces . 16 3 Deformation Theory for geometric objects 19 3.1 Functors of Artin rings . 19 3.2 Deformation theory of a compact complex manifold . 21 3.3 Deformation theory of a submanifold in a fixed complex man- ifold . 22 3.4 Deformation theory of a coherent sheaf on a fixed complex manifold . 23 3.5 Deformation theory of a pair consisting of a smooth divisor in a complex manifold . 23 3.6 Simultaneous deformations of a coherent sheaf and its com- plex base manifold . 25 3.7 Deformations of projective bundles . 26 3.8 Simultaneous deformations of a holomorphic vector bundle and a section . 29 4 Variation of Hodge structures on the cohomology of special complex manifolds 31 4.1 Variation of pure Hodge structures . 31 4.2 Logarithmic differentials and residues . 32 4.2.1 The complex of logarithmic differential forms . 32 4.2.2 The residue map . 33 4.2.3 The notion of mixed Hodge structures and variations of mixed Hodge structures . 41 4.2.4 The mixed Hodge structure on the hypercohomology of the complex of logarithmic differential forms . 42 4.2.5 Relative de Rham-Cohomology . 43 i ii CONTENTS 4.2.6 Hodge theory on quotients by finite groups . 45 5 Comparison between two deformation problems associated with a closed submanifold in a compact complex manifold 49 5.1 Statement of the main theorems . 49 5.2 Preparations . 50 5.3 Proof of Theorem 5.1.1 . 53 5.4 A generalization of Theorem 5.1.1 . 57 5.5 Proof of Theorem 5.1.2 . 59 5.6 An example for a potential function: the Noether-Lefschetz locus . 61 6 Simultaneous deformations of a holomorphic vector bundle and a section 65 6.1 Situation and main theorem . 65 6.2 Proof of Theorem 6.1.1 . 66 6.3 A special situation . 74 6.4 The structure of the sheaf Q ................... 78 6.5 A comparison theorem and a potential function . 84 7 Picard-Fuchs Equations for Calabi-Yau manifolds 91 7.1 Griffiths-Dwork reduction . 91 7.2 The Griffiths-Dwork reduction for families of hypersurfaces in projective spaces . 93 7.3 The Griffiths-Dwork reduction for complete intersections of codimesion 2 in a projective space . 97 7.3.1 Setup and computation of a residue map . 97 7.3.2 Pole order and Hodge filtration . 106 7.3.3 Proof of Theorem 7.3.9 . 108 7.3.4 Griffiths-Dwork reduction for codim-2 complete inter- sections . 119 7.3.5 Description of the method by Libgober and Teitel- baum for calculating the Picard-Fuchs equation . 125 8 Picard-Fuchs operators for pairs 131 8.1 Definition of residues for pairs . 132 8.2 Hodge and pole-order filtration . 137 8.3 A basis for relative cohomology . 139 8.4 The Griffiths-Dwork method for pairs . 140 8.4.1 The approach of Jockers and Soroush . 140 8.4.2 The work of Li, Lian and Yau . 143 8.4.3 Griffiths-Dwork reduction for pairs . 143 8.5 An example . 146 8.5.1 A family of Calabi-Yau 3-folds . 146 CONTENTS iii 8.5.2 A family of smooth divisors inside the family of Calabi- Yau 3-folds . 148 8.5.3 Picard-Fuchs operators for pX; Dq . 154 9 Picard-Fuchs operators for triples 159 9.1 A topological observation . 159 9.2 Definition of a cohomology for triples . 162 9.3 Pairing between homology and cohomology . 169 9.4 A residue map for triples . 173 9.5 Gauß -Manin connection . 173 9.6 An Example . 185 9.7 A modified example . 189 A Implementation of the algorithms as Singular programmes 191 A.1 A programme for computing the Picard-Fuchs equation of a complete intersection Calabi-Yau 3-fold . 191 A.2 A programme for calculating a Hodge number . 201 A.3 A programme for the computation of Picard-Fuchs operators of a pair consisting of a Calabi-Yau 3-fold and a divisor . 209 iv CONTENTS Chapter 1 Introduction Calabi-Yau manifolds and the general setting. Calabi-Yau manifolds are simply-connected compact projective manifolds X with trivial canonical n bundle KX “ ΩX , where n “ dim X, and the additional property that ´ ¯ 0 j H X; ΩX “ 0 for 2 ¤ j ¤ n ´ 1: By Yau's solution of the Calabi conjecture, X carries a K¨ahlermetric with vanishing Ricci curvature. Calabi-Yau manifolds are not only central objects in complex geometry, but also play an important role in string theory. In fact, much of the mathe- matical theory of Calabi-Yau manifolds has its origin in physics, e.g. mirror symmetry. Calabi-Yau manifolds themselves have a special property in de- formation theory: their first-order deformations are unobstructed, although 2 the obstructions live in a space H pX; TX q which is not 0. So their local 1 moduli are smooth of dimension dim H pX; TX q. Motivated by the study of D-branes in string theory, physicists started to study deformations of geometric objects on Calabi-Yau manifolds. For each joint moduli problem, e.g. the moduli problem of X together with a coherent sheaf on it, one has to specify the notion of a family of geometric objects. For example, a simultaneous deformation of a complex manifold X and a coherent sheaf F on X consists of a deformation X Ñ S of X, where X and S are complex spaces, and a coherent sheaf F~ on X , flat over S. These types of deformations are subject of this thesis. Picard-Fuchs equations and potential functions. Given a Calabi- Yau 3-fold X and a geometric object, say A, we are interested in the joint "local moduli space" M “ M pX; Aq of A and X. This local moduli space is realized as a subspace of the Zariski tangent space T – CN with N :“ T M M dim pX;Aq of at A, which is the space of first-order deformations. 1 2 CHAPTER 1. INTRODUCTION Thus, given a first-order deformation, one tries to extend it order by order. However in general there are obstructions to do this, and to understand how M sits in T is the same as to understand the obstructions. So M parametrizes { up to convergence { the deformations that are unobstructed to each order. The hope is then to have a so-called holomorphic potential function f : U Ñ C defined on an open set U Ă T such that the critical locus of f is M. More specifically, consider the space of first-order deformations T pX; Aq 1 of X and A, and the space of first-order deformations T pXq – H pX; TX q of X, leading to a forgetful morphism T pX; Aq Ñ T pXq. Then there should be open neighbourhoods U Ă T pX; Aq and V Ă T pXq of 0 and a holomorphic function f : U Ñ C such that ( M dU{V f “ 0 “ pX; Aq X U: (1.0.0.1) The technical tool to treat infinitesimal deformations are functors of Artin rings. These functors associate to an Artin ring the space of defor- mations of the given object over the spectrum of the Artin ring. In this setting, obstructions are cohomology classes that vanish if a deformation over the spectrum of an Artin ring can be extended to a deformation over the spectrum of a larger Artin ring. For geometric deformation problems, usually the space of first-order deformations is given by the first cohomology group (or Ext-group) of a certain coherent sheaf. The space of obstructions is contained in the second cohomology of the same coherent sheaf. In most cases, the obstructions of a deformation problem are difficult to compute. Therefore it is useful to have a potential function that gives information about the obstructions. Physicists propose that in some situations such a potential function should exist as a solution of a certain differential equation associated with the deformation problem, the Picard-Fuchs equation. In the special case of a family of Calabi-Yau manifolds, the Picard-Fuchs equation is satisfied by all periods induced by the variation of the complex structure that is related to the given deformation problem. Furthermore, the potential function should be a generating function of the obstructions. We explain this in more detail below. In several cases potential functions are known to exist, although not all expected properties are established yet. For example, sometimes just the existence of a holomorphic potential function is known, but it is still unknown whether it is a solution of a differential equation with specific properties. The existence problem for potential functions is therefore closely related to the task of constructing Picard-Fuchs equations associated with defor- 3 mation problems. These Picard-Fuchs equations are differential equations arising from certain Gauß-Manin connections. We explain this in the sim- plest possible situation. Let X Ñ T be a deformation of a Calabi-Yau n-fold over a complex manifold T , i.e., a family of Calabi-Yau manifolds, and let Ω be a fam- n ily of holomorphic n-forms on X .
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