Advanced Lectures in Mathematics Volume 42

International Presswww.intlpress.com

高 等 教 育 出 版 社HIGHER EDUCATION PRESS

edited by

Lizhen Ji · Shing-Tung Yau

Uniformization, Riemann-Hilbert Correspondence, Calabi-Yau Manifolds & Picard-Fuchs Equations

Copyright © 2018 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China. This work is published and sold in China exclusively by Higher Education Press of China. All rights reserved. Individual readers of this publication, and non-profit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given. Republication, systematic copying, or mass reproduction of any material in this publica-tion is permitted only under license from International Press. Excluded from these pro-visions is material in articles to which the author holds the copyright. (If the author holds copyright, notice of this will be given with the article.) In such cases, requests for per-mission to use or reprint should be addressed directly to the author. ISBN: 978-1-57146-363-0 Printed in the United States of America. 21 20 19 18 9 8 7 6 5 4 3 2 1

Advanced Lectures in Mathematics, Volume 42 Uniformization, Riemann-Hilbert Correspondence, Calabi-Yau Manifolds, and

Picard-Fuchs Equations Editors: Lizhen Ji (Department of Mathematics, University of Michigan) Shing-Tung Yau (Department of Mathematics, Harvard University)

ADVANCED LECTURES IN MATHEMATICS

Executive Editors

Editorial Board

Shing-Tung Yau Harvard University Cambridge, Mass., U.S.A. Lizhen Ji University of Michigan, Ann Arbor Ann Arbor, Mich., U.S.A.

Kefeng Liu University of California at Los Angeles Los Angeles, Calif., U.S.A. Zhejiang University Hangzhou, China

Chongqing Cheng Nanjing University Nanjing, China Zhong-Ci Shi Institute of Computational Mathematics Chinese Academy of Sciences (CAS) Beijing, China Zhouping Xin The Chinese University of Hong Kong Hong Kong, China Weiping Zhang Nankai University Tianjin, China Xiping Zhu Sun Yat-sen University Guangzhou, China

Tatsien Li Fudan University Shanghai, China Zhiying Wen Tsinghua University Beijing, China Lo Yang Institute of Mathematics Chinese Academy of Sciences (CAS) Beijing, China Xiangyu Zhou Institute of Mathematics Chinese Academy of Sciences (CAS) Beijing, China

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Preface

The uniformization theorem for Riemann surfaces is one of the most importantand beautiful theorems in mathematics. Its statement is clean, simple and natural,and it ties together several major areas of the main stream mathematics. The roadfrom the initial conception by Klein and Poincare to the final rigorous proofs byKoebe and Poincare of the uniformization theorem was filled with inspiration,competition, excitement and disappointment.

Poincare gave multiple proofs or attempts to establish different versions of theuniformization theorem. The work of Fuchs on differential equations with regularsingularities inspired Poincare’s earlier work on differential equations, automor-phic functions, and the uniformization theorem, which made him famous and alsopartially motivated Hilbert’s twenty-first problem on the existence of linear differ-ential equations with specified singular points and monodromy groups, which wasin turn generalized to the Riemann-Hilbert correspondence.

After the Riemann mapping theorem for domains in the complex plane andthe uniformization theorem for Riemann surfaces, many attempts were made toobtain higher dimensional generalizations. It seems that one of the most successfulgeneralizations is the notion of Kahler-Einstein metrics for compact complex man-ifolds. Calabi-Yau manifolds form a special class of Kahler-Einstein manifolds andhave many applications to subjects ranging from differential geometry, algebraicgeometry, mathematical physics, topology, number theory, algebra, etc. They arehigher dimensional generalizations of elliptic curves. Though many results areknown about Calabi-Yau manifolds, much more is waiting to be explored. Theperiods of elliptic curves satisfy the Picard-Fuchs equation, which is a special dif-ferential equation with regular singularities. Periods of Calabi-Yau manifolds alsosatisfy Picard-Fuchs equations.

In view of the interconnection of these topics, a conference titled “Uniformiza-tion, Riemann-Hilbert Correspondence, Calabi-Yau manifolds, and Picard-FuchsEquations” was held at Institute Mittag-Leffler in July 13–18, 2015. The purposeof the conference was to bring together many leading experts in these subjects toexplore their historical development and interconnection between them.

To keep a permanent record of this conference and also to continue the fruitfulinteraction between the participants, we decided to edit a book which can serveas an overview of the many topics discussed. Almost all papers of this book werecontributed by speakers of the conference. We hope and believe that they conveythe lively atmosphere and accessible style of the conference.

All participants agreed that this conference was held at one of the most pleas-ant places for mathematicians: the great mathematics library, views of the sur-roundings, and fresh fruits during the tea break. We are grateful to InstituteMittag-Leffler for their hospitality, to many speakers for their excellent talks and

ii Preface

contributions to this book, and to referees for their help in reviewing and improvingthe papers in this book.

Lizhen JiShing-Tung Yau

February 2017

Contents

Hypergeometric Functions, from Riemann till Present . . . . . . . . . . . . . . . . 1

Frits Beukers

Picard-Fuchs Uniformization of Modular Subvarieties . . . . . . . . . . . . . . 21

Brent Doran, Charles F. Doran, and Andrew Harder

Poincare’s Path to Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Connemara Doran

Exploring Modular Forms and the Cohomology of Local Systems

on Moduli Spaces by Counting Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Gerard van der Geer

Moduli Spaces of Compact Riemann Surfaces: Their Complex

Structures and an Overview of Major Results . . . . . . . . . . . . . . . . . 111

Lizhen Ji

The Morrison-Kawamata Cone Conjecture and Abundance on

Ricci Flat Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Vladimir Lazic, Keiji Oguiso, and Thomas Peternell

Conformal Geometry and the Painleve VI Equation . . . . . . . . . . . . . . . 187

Chang-Shou Lin

Calabi-Yau Metrics: Mirror Symmetry, Hitchin Systems and

Instanton Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Wenxuan Lu

Quasiconformal Mappings, from Ptolemy’s Geography to the

Work of Teichmuller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Athanase Papadopoulos

ii Contents

Riemann-Hilbert Correspondence, Irregular Singularities and

Hodge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Claude Sabbah

Krichever-Novikov Type Algebras and Wess-Zumino-Novikov-

Witten Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Martin Schlichenmaier

Moduli of Canonically Polarized Manifolds, Higher Order

Kodaira-Spencer Maps, and an Analogy to Calabi-Yau

Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Georg Schumacher

Calabi–Yau Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

Duco van Straten

A Survey on Toric Mirror Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

Hsian-Hua Tseng

Dynamical Systems on Analytic Manifolds of Quadratic

Differentials: Chapter I, F-Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

William A. Veech

Aspects on Calabi–Yau Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

Chin-Lung Wang

Degeneration of Ricci-flat Calabi-Yau Manifolds and Its

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551

Yuguang Zhang

Mirror Symmetry for Plane Cubics Revisited . . . . . . . . . . . . . . . . . . . . . . . 593

Jie Zhou

Hypergeometric Functions, from

Riemann till Present

Frits Beukers

Abstract

Hypergeometric functions form a family of classical functions that occur frequentlywithin many areas of mathematics and its applications. They were first introducedby Euler, who already discovered a great many surprising properties. Gauss con-tinued this study by considering hypergeometric functions as solutions of a secondorder differential equation in the complex plane, including their multivaluedness.Riemann took up Gauss’s study and made hypergeometric functions as prime ex-ample for his ideas on analytic continuation. It was also Riemann who namedthem Gauss hypergeometric functions. Although there exist many generalizationsnowadays, we concentrate ourselves on these original functions. We briefly sketchRiemann’s ideas and give an overview of developments around Gauss’s hypergeo-metric function until recent times. There is an overlap of the first two sections withthe author’s summer school notes in [1].

c© Higher Education Pressand International PressBeijing–Boston

Uniformization, Riemann-HilbertCorrespondence, Calabi-Yau Manifolds,and Picard-Fuchs EquationsALM 42, pp. 1–19

1 Definition, first properties

The hypergeometric function of Gauss (although Euler-Gauss might be more ap-propriate) is a function of one complex variable z and three parameters a, b, cwhich we take to be in R. Suppose that a, b, c ∈ R and c �∈ Z�0. Define Gauss’shypergeometric function by

2F1(a, b, c|z) =∑ (a)n(b)n

(c)nn!zn. (1.1)

The Pochhammer symbol (x)n is defined by (x)0 = 1 and (x)n = x(x + 1) · · · (x +n − 1). The radius of convergence of (1.1) is 1 unless a or b is a non-positiveinteger, in which cases we have a polynomial. Here are a couple of examples,