Progress in Mathematics Volume 249. Series Editors Hyman Bass Joseph Oesterle Alan Weinstein

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2 Progress in Mathematics Volume 249 Series Editors Hyman Bass Joseph Oesterle Alan Weinstein

3 Joachim Kock Israel Vainsencher An Invitation to Quantum Cohomology Kontsevich's Formula for Rational Plane Curves Birkhauser Boston Basel Berlin

4 Joachim Kock Israel Vainsencher Universitat Autonoma de Barcelona Universidade Federal de Minas Gerais Departament de Matematiques, Edifici C Departamento de Matematica ICEx Bellaterra (Barcelona) Belo Horizonte MG Spain Brazil Mathematics Subject Classification (2000): 14N35, 14N10 Library of Congress Control Number: ISBN e-isbn: ISBN Printed on acid-free paper Birkhauser Boston BivkhdUSer Based on the original Portuguese edition, A formula de Kontsevich para curvas racionais planas, Instituto de Matematica Pura e Aplicada, Rio de Janeiro, Brazil, 1999 J. Kock and I. Vainsencher All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights (TXQ/EB)

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6 Preface This book is an elementary introduction to some ideas and techniques that have revolutionized enumerative geometry: stable maps and quantum cohomology. A striking demonstration of the potential of these techniques is provided by Kontsevich's famous formula, which solves a long-standing question: How many plane rational curves of degree d pass through 3d 1 given points in general position? The formula expresses the number of curves for a given degree in terms of the numbers for lower degrees. A single initial datum is required for the recursion, namely, the case d = I, which simply amounts to the fact that through two points there is but one line. Assuming the existence of the Kontsevich spaces of stable maps and a few of their basic properties, we present a complete proof of the formula, and use the formula as a red thread in our Invitation to Quantum Cohomology. For more information about the mathematical content, see the Introduction. The canonical reference for this topic is the already classical Notes on Stable Maps and Quantum Cohomology by Fulton and Pandharipande [29], cited henceforth as FP-NOTES. We have traded greater generality for the sake of introducing some simplifications. We have also chosen not to include the technical details of the construction of the moduli space, favoring the exposition with many examples and heuristic discussions. We want to stress that this text is not intended as (and cannot be!) a substitute for FP-NOTES. Au contraire, we hope to motivate the reader to study the cited notes in depth. Have you got a copy? If not, point your browser to http: //arxiv. org/alg-geom/ , and getit at once. Prerequisites. We assume some basic algebraic geometry and some elementary intersection theory. For algebraic geometry: some familiarity with algebraic curves, divisors and line bundles, blowup, Grassmannians. Chapter 1 of Hartshorne [44] should be sufficient background, with some additional reading in Harris [42] for Grassmannians. We freely use the word "scheme" throughout, but do not make use of scheme theory in any essential way in fact, we hardly use any commutative

7 viii Preface algebra. Spending an evening with Eisenbud-Harris [22] may be sufficient background on schemes. For intersection theory we just need the notions of pullback and pushforward of cycles and classes, the intersection product, thefirstchem class of a line bundle, and Poincare duality. The standard reference for this material is Fulton [28]. The original Portuguese edition of this book was written to support a five-lecture mini-course given at the 22 Coloquio Brasileiro de Matematica and published by IMPA in The idea of the mini-course and the style of the exposition go back to another mini-course. Intersection theory over moduli spaces of curves [33], taught by Letterio Gatto in the Recife Summer School, in January He showed us that it was possible to explain stable maps in an intelligible way, and that Kontsevich's formula was not just unattainable magic from theoretical physics: indeed it constitutes material that fits perfectly in the venerable tradition of enumerative geometry. The text for the mini-course grew gradually from notes from seminars given by the first author upon three occasions in 1998: in Recife, Belo Horizonte, and Maragogi, Alagoas. This revised edition. Six years have past since the original Portuguese edition of this book appeared, and the subject of Gromov-Witten theory has evolved considerably. Speakers at conferences can nowadays say stable map with the same aplomb as six years ago they could say stable curve, assuming that the audience knows the definition, more or less. While the audience is getting used to the words, the magic surrounding the basics of the subject is still there, for better or for worse, both as fascinating mathematics, and sometimes as secret conjuration. However, for the student who wishes to get into the subject, the learning curve of FP-NOTES can appear quite steep. We feel there is still a need for a more elementary text on these matters, perhaps even more today, due to the rapid expansion of the subject. We hope this English translation can help in filling this gap. This is a revised and expanded translation. Some errors have been corrected, some sections have been reorganized, and some clarifications of subtler points have been added. A short prologue with a few explicit statements on cross ratios and with a brief account of moduli spaces has been included; in Chapter 5 we have added a quick primer on generating functions. The five sections entitled "Generalizations and references" have been expanded, many more exercises have been included, and the Bibliography has been updated. Acknowledgements, We thank the organizers of the 22 Coloquio Brasileiro de Matematica for the opportunity to give the minicourse, and the audience for their precious feedback. We are much indebted to Letterio Gatto for revealing the secrets of quantum cohomology to us, and for his support all along. Thanks are also due to Elizabeth Gasparim, Francesco Russo, and in particular Ty Le Tat, for reading and

8 Preface ix commenting on preliminary versions, and we are grateful to Steven Kleiman, Rahul Pandharipande, Aaron Bertram, Barbara Fantechi, and Anders Kock for kindly answering some questions related to the text. The first (resp. second) author was supported by a grant from the Danish Natural Science Research Council (resp. Brazil's CNPq) and registers here his gratitude. Joachim Kock and Israel Vainsencher Recife, April 1999 Barcelona & Belo Horizonte, August 2005

9 Contents Preface vii Introduction 1 0 Prologue: Warming up with Cross Ratios, and the Definition of Moduli Space Cross ratios Definition of moduli space 11 1 Stable ^-pointed Curves /I-pointed smooth rational curves Stable/I-pointed rational curves Stabilization, forgetting marks, contraction Sketchof the construction of Mo,^ The boundary Generalizations and references 39 2 Stable Maps Maps F^ ^ ' parameter families Kontsevich stable maps Idea ofthe construction of Mo,«(lP'',^) Evaluation maps Forgetful maps The boundary Easy properties and examples Complete conies Generalizations and references 78 3 Enumerative Geometry via Stable Maps Classical enumerative geometry Counting conies and rational cubics via stable maps... 95

10 xii Contents 3.3 Kontsevich's formula Transversality and enumerative significance Stable maps versus rational curves Generalizations and references Gromov-Witten Invariants Definition and enumerative interpretation Ill 4.2 Properties of Gromov-Witten invariants Recursion The reconstruction theorem Generalizations and references Quantum Cohomology Quick primer on generating functions The Gromov-Witten potential and the quantum product Associativity Kontsevich's formula via quantum cohomology Generalizations and references 141 Bibliography 149 Index 157

11 An Invitation to Quantum Cohomology