Aspects of Mathematics Friedrich Hirzebruch · Thomas Berger Rainer Jung Manifolds and Modular Forms A Publication of the Max-Planck-Institut für Mathematik, Bonn Second Edition Friedrich Hirzebruch Thomas Berger Rainer jung Manifolds and Modular Forms Aspect~f tv\athematic~ Edited by Klas Diederich Vol. E 2: M. Knebusch/M. Kolster: Wittrings Vol. E 3: G. Hector/U. Hirsch: Introduction to the Geometry of Foliations, Part B Vol. E 5: P. Stiller: Automorphic Forms and the Picard Number of an Elliptic Surface Vol. E 6: G. Faltings/G. Wustholz et al.: Rational Points* Vol. E 7: W. Stoll: Value Distribution Theory for Meromorphic Maps Vol. E 9: A Howard/P.-M. Wong (Eds.): Contribution to Several Complex Variables Vol. E 10: A J. Tromba (Ed.): Seminar of New Results in Nonlinear Partial Differential Equations* Vol. E 13: Y. Andre: G-Functions and Geometry* Vol. E 14: U. Cegrell: Capacities in Complex Analysis Vol. E 15: J.-P. Serre: lectures on the Mordeii-Weil Theorem Vol. E 16: K. lwasaki/H. Kimura/S. Shimomura/M. Yoshida: From Gauss to Painleve Vol. E 17: K. Diederich (Ed.): Complex Analysis Vol. E 18: W. W. J. Hulsbergen: Conjectures in Arithmetic Algebraic Geometry Vol. E 19: R. Rocke: lectures on Nonlinear Evolution Equations Vol. E 20: F. Hirzebruch, Th. Berger, R. Jung: Manifolds and Modular Forms* Vol. E 21: H. Fujimoto: Value Distribution Theory of the Gauss Map of Minimal Surfaces in Rm Vol. E 22: D. V. Anosov/A A Bolibruch: The Riemann-Hilbert Problem Vol. E 23: A P. Fordy/J. C. Wood (Eds.): Harmonic Maps and Integrable Systems Vol. E 24: D. S. Alexander: A History of Complex Dynamics Vol. E 25: A Tikhomirov/A Tyurin (Eds.): Algebraic Geometry and its Applications *A Publication of the Max·Pianck·lnstitut fur Mathematik, Bonn Volumes of the German-language subseries "Aspekte der Mathematik" ore listed at the end of the book. Friedrich Hirzebruch Thomas Berger Rainer jung Manifolds and Modular Forms Translated by PeterS. Landweber A Publication from the Max-Pianck-lnstitut fur Mathematik, Bonn Second Edition I I VJeweg Professor Dr. Friedrich Hirzebruch, Thomas Berger, and Rainer Jung Max-Planck-Institut ftir Mathematik Gottfried-Claren-Str. 26 53225 Bonn Germany First Edition 1992 Second Edition 1994 Appendix III: Elliptic genera of level N for complex manifolds reprinted with permission of Kluwer Academic Publishers Mathematics Subject Classification: 57-02, llFll, 33C45, 33E05, 55N22, 55R10, 57R20, 58G10 Ali rights reserved © Springer Fachmedien Wiesbaden 1994 Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, in 1994 No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise without prior permission of the copyright holder. Cover design: Wolfgang Nieger, Wiesbaden Typeset using ArborText Publisher and TEX ISSN 0179-2156 ISBN 978-3-528-16414-0 ISBN 978-3-663-10726-2 (eBook) DOI 10.1007/978-3-663-10726-2 Preface During the winter term 1987/88 I gave a course at the University of Bonn under the title "Manifolds and Modular Forms". I wanted to develop the theory of "Elliptic Genera" and to learn it myself on this occasion. This theory due to Ochanine, Landweber, Stong and others was relatively new at the time. The word "genus" is meant in the sense of my book "Neue Topologische Methoden in der Algebraischen Geometrie" published in 1956: A genus is a homomorphism of the Thorn cobordism ring of oriented compact manifolds into the complex numbers. Fundamental examples are the signature and the A-genus. The A-genus equals the arithmetic genus of an algebraic manifold, provided the first Chern class of the manifold vanishes. According to Atiyah and Singer it is the index of the Dirac operator on a compact Riemannian manifold with spin structure. The elliptic genera depend on a parameter. For special values of the parameter one obtains the signature and the A-genus. Indeed, the universal elliptic genus can be regarded as a modular form with respect to the subgroup r0(2) of the modular group; the two cusps giving the signature and the A-genus. Witten and other physicists have given motivations for the elliptic genus by theoretical physics using the free loop space of a manifold. This led Witten to conjectures concerning the rigidity of the elliptic genus generalizing a result of Atiyah and myself on the rigidity of the equivariant A-genus: If the circle 51 acts on a spin manifold, then the equivariant .A-genus is a formal difference of two finite dimensional vector spaces (positive and negative harmonic spinors) which are representation spaces of 51. Our theorem (published in 1970) states that the difference of the characters of these representations (a finite Laurent series) is a constant, indeed equal to zero if the action is non-trivial. The proof uses the Atiyah-Singer equivariant index and fixed point theorem applied to the 51-action. The rigidity theorem in the case of the equivariant universal elliptic genus of spin manifolds with 51-action was first proved by Taubes and then by Bott and Taubes. I did not know it during the time of my course. I stressed special cases (strict multiplicativity in certain fibre bundles) following the pioneers Ochanine and Landweber. Shortly after my course the proceedings of a Princeton meeting (1986) edited by Landwe ber became available. They give an excellent description of the history of the young theory and the state of the art around the time of my course. During the second part of the course (changing the original program) I began to develop the theory of elliptic genera of level N for complex manifolds, finding out that the differential equations for the characteristic functions are given by polynomials which I called almost-Chebyshev. These are polynomials such that the critical values except one are all equal up to sign. A little later I learned from Th. J. Rivlin that such polynomials were already well-known under the name Zolotarev polynomials. The universal elliptic VI Preface genus of level N is a modular form for r 1 (N), the values in the cusps being certain holomorphic Euler numbers (if the first Chern class of the manifold is divisible by N ). In this way the universal elliptic genus of level N is very close to the Riemann-Roch theorem formulated and proved in my book mentioned above. During the winter term 1987/88 I was occasionally away on official business. My course was taken over several times by N.-P. Skoruppa who gave a thorough and expert presentation ofresults on modular forms needed for the theory. P. Baum was visiting the Max-Planck-Institut. He also lectured in my course and gave an introduction to the Dirac operator. In this way the first two of the four appendices of this book came into existence. The course was directed to students after their Vordiplom. Some of them had basic knowledge on characteristic classes. However, I tried to explain without proofs all the prerequisite material to make the course understandable to a large audience: characteristic classes, cobordism theory, Atiyah-Singer index theorem, Riemann-Roch theorem, etc. Thomas Berger and Rainer Jung belonged to the students and prepared notes. This was a considerable job, because I did not have a usable manuscript. The result of their efforts were inofficial lecture notes in German which became ready in July 1988. P. Landweber used these notes for a course at Rutgers University in the spring of 1989 and prepared a translation into English for his students; in addition he proposed numerous corrections and improvements. Thus the present English text materialized. For the German version of the notes also Gottfried Barthel and Michael Puschnigg contributed improvements. The book has two further appendices: In August 1987 the sixteenth of the well-known and highly estimated International Conferences on Differential Geometrical Methods in Theoretical Physics organized by Konrad Bleuler took place in Como. I was invited to write up my talk for the proceedings. The talk was a general survey on elliptic genera. Instead of writing this up, I wrote a paper "Elliptic genera of level N for complex manifolds" which is reproduced here as Appendix III, incorporating some corrections and improvements by Thomas Berger. A rigidity theorem for the level N genus is contained in the paper. It was proved during my visit in Cambridge (England) in March 1988 when I was a guest of Robinson College. When Michael Atiyah came to Bonn in February 1988 he explained to me Bott's approach to the rigidity for spin manifolds mentioned above and indicated its relationship to our old paper on the A-genus. (The Atiyah-Singer equivariant index and fixed point theorem is used.) We had further discussions in Oxford in March 1988, before I proved the result for the level N case (consulting Bott's Cargese lectures). Later the paper by Bott and Taubes became available. The role of the Zolotarev polynomials was already pointed out. At the end of my paper I announced the plan to write a separate paper on these polynomials and their role for the elliptic genus. But, indeed, Rainer Jung took over. This became the topic of his Diplomarbeit, in which he carried out my plan and proved several other interesting theorems. His Appendix IV is a condensed version of the Diplomarbeit. Preface vii I wish to thank all the mathematicians mentioned above for their cooperation and help. I want to express special thanks to my coauthors Thomas Berger and Rainer Jung who wrote the original notes and to whom the book owes its existence. Many thanks are due to Peter Landweber for the translation and for all mathematical help. I am grateful that Nils-Peter Skoruppa and Paul Baum added their lectures to the book, and I must thank again Rainer Jung for adding his Diplomarbeit and for much work during the final stages of the preparation of the manuscript. Many thanks are due also to Mrs. Iris Abdel Hafez of the Max-Planck-Institut for her excellent work in typesetting the manuscript. Bonn F. Hirzebruch March 23, 1992 Preface to the second edition The text remained almost unchanged. Only some slight obscurities and several misprints have been corrected. The authors would like to express thanks to all readers who have contributed their comments or will do so in the future. We are also indebted to the computing centre of the Max-Planck-Institut managed by Sven Maurmann for support during the preparation of both editions of this book. Bonn F. Hirzebruch September 1993 Th. Berger R. Jung Notice to the reader The special sign "diJ" after a proof indicates the end of that proof. If it occurs at the end of a stated theorem, proposition, lemma or corollary no proof for that statement will be given. Contents Preface . ............................... v Chapter 1 Background I Cobordism Theory 2 Characteristic classes 2 3 Pontrjagin classes of quaternionic projective spaces . . . 5 4 Characteristic classes and invariants 7 5 Representations and vector bundles . 9 6 Multiplicative sequences and genera . 13 7 Calculation of <p( Pk (!HI)) . 15 8 Complex genera .. . ................... 18 Chapter 2 Elliptic genera The WeierstraB p-function . . 23 2 Construction of elliptic genera .. . . 25 3 An excursion on the lemniscate .. . ..... 29 4 Geometric complement on the addition theorem . . . . . . . .. 32 Chapter 3 A universal addition theorem for genera Virtual submanifolds . 35 2 A universal genus . 38 Chapter 4 Multiplicativity in fibre bundles I The signature and the £-genus ........ . 41 2 Algebraic preliminaries .. . 44 3 Topological preliminaries . 46 4 The splitting principle .. . 48 5 Integration over the fibre . ...... . . 51 6 Multiplicativity and strict multiplicativity ...... .. 52 x Contents Chapter 5 The Atiyah-Singer index theorem Elliptic operators and elliptic complexes .. . .. 57 2 The index of an elliptic complex . 59 3 The de Rham complex . 59 4 The Dolbeault complex . . . 60 5 The signature as an index .. 63 6 The equivariant index ... ......... . 66 7 The equivariant Xy-genus for S1-actions . 68 8 The equivariant signature for S1-actions ...... ... 71 Chapter 6 Twisted operators and genera Motivation for elliptic genera after Ed Witten . 73 2 The expansion at the cusp 0 .77 3 The Witten genus . . . . . . . . 82 4 The Witten genus and the Lie group E8 . • ... 89 5 Plumbing of manifolds ... 92 Chapter 7 Riemann-Roch and elliptic genera in the complex case Elliptic genera of level N . 97 2 The values at the cusps .. . 98 3 The equivariant case and multiplicativity .. 100 4 The loop space and the expansion at a cusp . . . . . . . . . . 102 5 The differential equation 103 6 The modular curve . . . 107 Chapter 8 A divisibility theorem for elliptic genera The theorem of Ochanine .............. . 113 2 Proof of Ochanine's theorem ................... 117