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

Advanced Lectures in Mathematics Volume 42

edited by Lizhen Ji · Shing-Tung Yau

International Press 高等教育出版社 www.intlpress.com HIGHER EDUCATION PRESS 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)

This work is published and sold in China exclusively by Higher Education Press of China.

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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

Executive Editors

Shing-Tung Yau Kefeng Liu Harvard University University of California at Los Angeles Cambridge, Mass., U.S.A. Los Angeles, Calif., U.S.A. Zhejiang University Lizhen Ji Hangzhou, China University of Michigan, Ann Arbor Ann Arbor, Mich., U.S.A.

Editorial Board

Chongqing Cheng Tatsien Li Nanjing University Fudan University Nanjing, China Shanghai, China

Zhong-Ci Shi Zhiying Wen Institute of Computational Mathematics Tsinghua University Chinese Academy of Sciences (CAS) Beijing, China Beijing, China

Zhouping Xin Lo Yang The Chinese University of Hong Kong Institute of Mathematics Hong Kong, China Chinese Academy of Sciences (CAS) Beijing, China

Weiping Zhang Xiangyu Zhou Nankai University Institute of Mathematics Tianjin, China Chinese Academy of Sciences (CAS) Beijing, China Xiping Zhu Sun Yat-sen University Guangzhou, China Participants in the conference “Uniformization, Riemann-Hilbert Correspondence, Calabi-Yau manifolds, and Picard-Fuchs Equations” Institute Mittag-Leffler, July 2015 Preface

The uniformization theorem for Riemann surfaces is one of the most important and beautiful theorems in mathematics. Its statement is clean, simple and natural, and it ties together several major areas of the main stream mathematics. The road from the initial conception by Klein and Poincaré to the final rigorous proofs by Koebe and Poincaré of the uniformization theorem was filled with inspiration, competition, excitement and disappointment. Poincaré gave multiple proofs or attempts to establish different versions of the uniformization theorem. The work of Fuchs on differential equations with regular singularities inspired Poincaré’s earlier work on differential equations, automor- phic functions, and the uniformization theorem, which made him famous and also partially motivated Hilbert’s twenty-first problem on the existence of linear differential equations with specified singular points and monodromy groups, which was in turn generalized to the Riemann-Hilbert correspondence. After the Riemann mapping theorem for domains in the complex plane and the uniformization theorem for Riemann surfaces, many attempts were made to obtain higher dimensional generalizations. It seems that one of the most successful generalizations is the notion of Kähler-Einstein metrics for compact complex manifolds. Calabi-Yau manifolds form a special class of Kähler-Einstein manifolds and have many applications to subjects ranging from differential geometry, algebraic geometry, mathematical physics, topology, number theory, algebra, etc. They are higher dimensional generalizations of elliptic curves. Though many results are known about Calabi-Yau manifolds, much more is waiting to be explored. The periods of elliptic curves satisfy the Picard-Fuchs equation, which is a special differential equation with regular singularities. Periods of Calabi-Yau manifolds also satisfy Picard-Fuchs equations. In view of the interconnection of these topics, a conference titled “Uniformiza- tion, Riemann-Hilbert Correspondence, Calabi-Yau manifolds, and Picard-Fuchs Equations” was held at Institute Mittag-Leffler in July 13–18, 2015. The purpose of the conference was to bring together many leading experts in these subjects to explore their historical development and interconnection between them. To keep a permanent record of this conference and also to continue the fruitful interaction between the participants, we decided to edit a book which can serve as an overview of the many topics discussed. Almost all papers of this book were contributed by speakers of the conference. We hope and believe that they convey the 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 Institute Mittag-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 improving the papers in this book.

Lizhen Ji Shing-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

Poincar´e’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 Lazi´c, Keiji Oguiso, and Thomas Peternell

Conformal Geometry and the Painlev´eVIEquation ...... 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 Uniformization, Riemann-Hilbert c Higher Education Press Correspondence, Calabi-Yau Manifolds, and International Press and Picard-Fuchs Equations Beijing–Boston ALM 42, pp. 1–19

Hypergeometric Functions, from Riemann till Present

Frits Beukers

Abstract

Hypergeometric functions form a family of classical functions that occur frequently within many areas of mathematics and its applications. They were first introduced by Euler, who already discovered a great many surprising properties. Gauss con- tinued this study by considering hypergeometric functions as solutions of a second order 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 named them Gauss hypergeometric functions. Although there exist many generalizations nowadays, we concentrate ourselves on these original functions. We briefly sketch Riemann’s ideas and give an overview of developments around Gauss’s hypergeometric function until recent times. There is an overlap of the first two sections with the author’s summer school notes in [1].

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, c whichwetaketobeinR. Suppose that a, b, c ∈ R and c ∈ Z0. Deﬁne Gauss’s hypergeometric function by (a)n(b)n n 2F1(a, b, c|z)= z . (1.1) (c)nn!

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