An Invitation to Quantum Cohomology: Kontsevich's Formula for Rational Plane Curves (Progress in Mathematics (249))

This book is an elementary introduction to stable maps and quantum cohomology, starting with an introduction to stable pointed curves, and culminating with a proof of the associativity of the quantum product. The viewpoint is mostly that of enumerative geometry, and the red thread of the exposition is the problem of counting rational plane curves. Kontsevich's formula is initially established in the framework of classical enumerative geometry, then as a statement about reconstruction for Gromov–Witten invariants, and finally, using generating functions, as a special case of the associativity of the quantum product. Emphasis is given throughout the exposition to examples, heuristic discussions, and simple applications of the basic tools to best convey the intuition behind the subject. The book demystifies these new quantum techniques by showing how they fit into classical algebraic geometry. Some familiarity with basic algebraic geometry and elementary intersection theory is assumed. Each chapter concludes with some historical comments and an outline of key topics and themes as a guide for further study, followed by a collection of exercises that complement the material covered and reinforce computational skills. As such, the book is ideal for self-study, as a text for a mini-course in quantum cohomology, or as a special topics text in a standard course in intersection theory. The book will prove equally useful to graduate students in the classroom setting as to researchers in geometry and physics who wish to learn about the subject. Front cover......Page 1 Series......Page 2 Title page......Page 3 Date-line......Page 4 Dedication......Page 5 Preface......Page 7 Contents......Page 11 Title......Page 13 Introduction......Page 15 0.1 Cross ratios......Page 19 0.2 Definition of moduli space......Page 25 1.1 $n$-pointed smooth rational curves......Page 35 1.2 Stable $n$-pointed rational curves......Page 37 1.3 Stabilization, forgetting marks, contraction......Page 42 1.4 Sketch of the construction of $\\bar{M}_{o,n}$......Page 46 1.5 The boundary......Page 48 1.6 Generalizations and references......Page 53 2.1 Maps $\\mathbb{P}^1 \\to \\mathbb{P}^r......Page 61 2.2 1-parameter families......Page 68 2.3 Kontsevich stable maps......Page 72 2.4 Idea of the construction of $\\bar{M}_{o,n}(\\mathbb{P}^r,d)$......Page 74 2.5 Evaluation maps......Page 77 2.6 Forgetful maps......Page 79 2.7 The boundary......Page 83 2.8 Easy properties and examples......Page 85 2.9 Complete conies......Page 88 2.10 Generalizations and references......Page 92 3.1 Classical enumerative geometry......Page 105 3.2 Counting conies and rational cubics via stable maps......Page 109 3.3 Kontsevich's formula......Page 113 3.4 Transversality and enumerative significance......Page 114 3.5 Stable maps versus rational curves......Page 116 3.6 Generalizations and references......Page 120 4.1 Definition and enumerative interpretation......Page 125 4.2 Properties of Gromov-Witten invariants......Page 129 4.3 Recursion......Page 131 4.4 The reconstruction theorem......Page 134 4.5 Generalizations and references......Page 137 5.1 Quick primer on generating functions......Page 143 5.2 The Gromov-Witten potential and the quantum product......Page 146 5.3 Associativity......Page 150 5.4 Kontsevich's formula via quantum cohomology......Page 152 5.5 Generalizations and references......Page 155 Bibliography......Page 163 Index......Page 171 Back cover......Page 174 "This book is an elementary introduction to stable maps and quantum cohomology, starting with an introduction to stable pointed curves, and culminating with a proof of the associativity of the quantum product. The viewpoint is mostly that of enumerative geometry, and the red thread of the exposition is the problem of counting rational plane curves. Kontsevich's formula in initially established in the framework of classical enumerative geometry, then as a statement about reconstruction for Gromov-Witten invariants, and finally, using generating functions, as a special case of the associativity of the quantum product." "Emphasis is given throughout the exposition of examples, heuristic discussions, and simple applications of the basic tools to best convey the intuition behind the subject. The book demystifies these new quantum techniques by showing how they fit into classical algebraic geometry." "Some familiarity with basic algebraic geometry and elementary intersection theory is assumed. Each chapter concludes with some historical comments and an outline to key topics and themes as a guide for further study, followed by a collection of exercises that complement the material covered and reinforce computational skills. As such, the book is ideal for self-study, as a text for a mini-course in quantum cohomology, or as a special topics text in a standard course in intersection theory. The book will prove equally useful to graduate students in the classroom setting as to researchers in geometry and physics who wish to learn about the subject."--Jacket 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 Kont- vich'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. Elementary introduction to stable maps and quantum cohomology presents the problem of counting rational plane curves Viewpoint is mostly that of enumerative geometry Emphasis is on examples, heuristic discussions, and simple applications to best convey the intuition behind the subject Ideal for self-study, for a mini-course in quantum cohomology, or as a special topics text in a standard course in intersection theory Provides introduction to stable maps and quantum cohomology that presents the problem of counting rational plane curves. This title is suitable for self-study, for a mini-course in quantum cohomology, or as a special topics text in a standard course in intersection theory.