An Introduction to Families, Deformations and Moduli

241 Pages·2010·3.445 MB·English

by  T. E. Venkata Balaji

Moduli Theory is one of those areas of Mathematics that has fascinated minds from classical T. E. Venkata Balaji
to modern times. This has been so because it reveals beautiful Geometry naturally hidden
in questions involving classification of geometric objects and because of the profound use
of the methods of several areas of Mathematics like Algebra, Number Theory, Topology and
Analysis to achieve this revelation. A study of Moduli Theory would therefore give senior An Introduction to
undergraduate and graduate students an integrated view of Mathematics. The present book
is a humble introduction to some aspects of Moduli Theory. Families, Deformations
and Moduli
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ISBN: 978-3-941875-32-6 Universitätsdrucke Göttingen Universitätsdrucke Göttingen

T. E. Venkata Balaji
An Introduction to Families, Deformations and Moduli

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erschienen in der Reihe der Universitätsdrucke
im Universitätsverlag Göttingen 2010
T. E. Venkata Balaji

An Introduction
to Families,
Deformations and
Moduli

Universitätsverlag Göttingen
2010
Bibliographische Information der Deutschen Nationalbibliothek
Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen
Nationalbibliographie; detaillierte bibliographische Daten sind im Internet über
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Address of the Author
T. E. Venkata Balaji
Department of Mathematics
Indian Institute of Technology Madras
Chennai 600036, India.
e-mail: [email protected]

This work is protected by German Intellectual Property Right Law.
It is also available as an Open Access version through the publisher’s homepage and
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read, download and distribute it. Users may also print a small number for educational
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Satz und Layout: T. E. Venkata Balaji
Umschlaggestaltung: Jutta Pabst
Titelabbildung: T. E. Venkata Raghavan

© 2010 Universitätsverlag Göttingen
http://univerlag.uni-goettingen.de
ISBN: 978-3-941875-32-6
to
my father
Shri T. E. Parthasarathy

Contents
Preface ix
Introduction xi
The Goals of this Book xxi
A Historical Note xxiii
1 Classification of Annuli and Elliptic Curves 1
1.1 Overview of this Chapter . . . . . . . . . . . . . . . . . . . 1
1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Topological Coverings . . . . . . . . . . . . . . . . . 2
1.2.2 Branched and Unbranched Coverings of Riemann
Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Relations betweenπππ (((MMM))) and Coverings ofMMM . . . 6
111
1.3 Uniformization of Riemann Surfaces . . . . . . . . . . . . . 8
1.3.1 The Fundamental Theorem . . . . . . . . . . . . . . 9
1.3.2 Surfaces with Universal Covering the Sphere . . . . 10
1.3.3 Surfaces with Universal CoveringCCC. . . . . . . . . . 10
1.3.4 Surfaces with abelianπππ and CoveringUUU . . . . . . 11
111
1.4 Classification of Annuli up to Conformal Equivalence . . . . 15
1.5 Classification of Elliptic Curves . . . . . . . . . . . . . . . . 17
1.5.1 Set-theoretic Classification of Elliptic Curves . . . . 17
1.5.2 Quotients, Projective Embeddings and Automorphic
Functions . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.3 The Riemann Surface Structure onUUU///PPPSSSLLL(((222,,,ZZZ))) . . 30
2 Families: Global Deformations 33
2.1 Overview of this Chapter . . . . . . . . . . . . . . . . . . . 33
2.2 Differentiable Families . . . . . . . . . . . . . . . . . . . . . 34
2.2.1 The Definition of a Differentiable Family. . . . . . . 34
iii
iv CONTENTS
2.2.2 Examples of Differentiable Families . . . . . . . . . . 36
2.2.3 NotionsofTrivialityandOperationsonDifferentiable
Families . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 The Fundamental Idea of Kodaira-Spencer . . . . . . . . . . 40
2.3.1 The Local Triviality of Differentiable Families . . . . 40
2.3.2 CCC∞∞∞-Deformations of Complex Structure . . . . . . . 40
2.4 Complex Analytic Families . . . . . . . . . . . . . . . . . . 45
2.4.1 The Definition of a Complex Analytic Family . . . . 46
2.4.2 Examples of Complex Analytic Families . . . . . . . 48
2.4.3 Notions of Triviality and Operations on Complex
Analytic Families . . . . . . . . . . . . . . . . . . . . 48
2.4.4 Remarks on Holomorphic Deformations of Complex
Structure . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Functorial Properties of Families . . . . . . . . . . . . . . . 51
2.5.1 The Functor of Families . . . . . . . . . . . . . . . . 51
2.5.2 The Functor of Equivalence Classes of Families . . . 53
2.6 Two Motivating Examples of Families of Complex Tori . . . 53
2.6.1 The Complex Analytic FamilyBBB of Complex Tori . 53
2.6.2 The Complex Analytic FamilyCCC of Complex Tori . . 54
2.7 Algebraizability and Analytic Deformations . . . . . . . . . 57
2.7.1 Algebraizability of Complex Tori . . . . . . . . . . . 58
2.7.2 Non-algebraic Deformations of Complex Algebraic
Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.8 Discontinuous/Continuous Variation of Complex Structure. 60
2.8.1 Continuous Variation of Complex Structure . . . . . 60
2.8.2 DiscontinuousVariationofComplexStructure: Jump
Phenomena . . . . . . . . . . . . . . . . . . . . . . . 60
3 Theory of Local Moduli: Infinitesimal Deformations 63
3.1 Overview of this Chapter . . . . . . . . . . . . . . . . . . . 63
3.2 InfinitesimalDeformations&DeformationMapsofKodaira-
Spencer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.1 Infinitesimal Deformations for Differentiable
Families . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.2 InfinitesimalKodaira-SpencerMapsforDifferentiable
Families . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2.3 The Fundamental Sequence of Vector Bundles for a
Differentiable Family . . . . . . . . . . . . . . . . . . 69
3.2.4 Reformulation of the Definition of Differentiable
Family in Terms of Differentiable Fiber Bundles . . 70
3.2.5 The Fundamental Sequence of Sheaves for a
Differentiable Family . . . . . . . . . . . . . . . . . . 71