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The handle http://hdl.handle.net/1887/44549 holds various files of this Leiden University dissertation.
Author: Zhao, Y.
Title: Deformations of nodal surfaces Issue Date: 2016-12-01
Deformations of nodal surfaces
Proefschrift ter verkrijging van
de graad van Doctor aan de Universiteit Leiden op gezag van Rector Magnificus prof. mr. C.J.J.M. Stolker,
volgens besluit van het College voor Promoties te verdedigen op 1 december 2016
klokke 16:15 uur
door
Yan Zhao
geboren te Fuzhou, China, in 1988
Promotor: Prof. dr. Peter Stevenhagen
Promotor: Prof. dr. Lambertus van Geemen (Universit`a degli studi di Milano)
Copromotor: Dr. Ronald M. van Luijk
Samenstelling van de promotiecommissie:
Prof. dr. Sebastiaan J. Edixhoven
Prof. dr. Chris A. M. Peters (Technische Universiteit Eindhoven / Universit´e Grenoble Alpes)
Prof. dr. Joseph H. M. Steenbrink (Radboud Universiteit Nijmegen) Prof. dr. Jaap Top (Rijksuniversiteit Groningen)
Prof. dr. Adrianus W. van der Vaart
This work was funded by Erasmus Mundus Algant-Doc and was carried out at Universiteit Leiden and Universit`a degli studi di Milano.
Contents
1 Introduction 1
1.1 Background . . . . 1
1.1.1 Geometric realization of Hodge structures . . . . 2
1.1.2 Deformations and Torelli type results . . . . 4
1.2 Organization of the thesis . . . . 6
2 Hodge theory 9 2.1 Review . . . . 9
2.2 V-manifolds . . . . 13
2.2.1 Definition and first properties . . . . 13
2.2.2 Tangent sheaves . . . . 15
2.2.3 V-manifolds as divisors on smooth varieties . . . . 17
2.2.4 Example: nodal surfaces . . . . 18
2.3 Hodge structure of singular hypersurfaces . . . . 23
3 Deformation theory 29 3.1 The Kodaira-Spencer map . . . . 29
3.1.1 Kodaira-Spencer map for divisors on varieties . . . . 31
3.1.2 Kodaira-Spencer map for quotient varieties . . . . 33
3.2 Infinitesimal period map . . . . 36
3.3 Infinitesimal Torelli theorem for nodal surfaces . . . . 38
4 Nodal sextic surfaces 49 4.1 56-nodal sextic surfaces . . . . 51
4.1.1 Construction of a family of even 56-nodal surfaces . . . 51
4.1.2 Coverings of Θ . . . . 54
4.1.3 Deformations of even 56-nodal surfaces . . . . 56
4.1.4 Construction of explicit examples . . . . 62
4.2 40-nodal sextic surfaces . . . . 64
4.2.1 Gallarati’s construction . . . . 65
4.2.2 Universality of Gallarati’s construction . . . . 75
4.2.3 The Casnati-Catanese construction . . . . 78
4.2.4 EPW sextics . . . . 80
4.2.5 Involutions on certain EPW sextic surfaces . . . . 86
5 Generalizations using mixed Hodge modules 91 5.1 Perverse sheaves . . . . 91
5.2 Mixed Hodge Modules . . . . 94
5.3 Hodge theory of singular varieties . . . . 98
Bibliography 103
Acknowledgements 109
Summary 111
Samenvatting 113
Sommario 115
Curriculum vitae 117
