An Invitation to Quantum Cohomology

1. Front Matter

   Pages i-xiii

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2. Introduction

   Pages 1-4

3. Prologue: Warming Up with Cross Ratios, and the Definition of Moduli Space

   Pages 5-19

4. Stable n-pointed Curves

   Pages 21-45

5. Stable Maps

   Pages 47-90

6. Enumerative Geometry via Stable Maps

   Pages 91-109

7. Gromov—Witten Invariants

   Pages 111-128

8. Quantum Cohomology

   Pages 129-148

9. Back Matter

   Pages 149-159

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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.

• Grad
• algebraic geometry
• cohomology
• homology
• moduli space

"The book seems to be ideally designed for a semester course or ambitious self-study."  —Mathematical Reviews

"The book is intended to be a friendly introduction to quantum cohomology. It makes the reader acquainted with the notions of stable curves and stable maps, and their moduli spaces. These notions are central in the field. ... Each chapter ends with references for further readings, and also with a set of exercices which help fixing the ideas introduced in that chapter. This makes the book especially useful for graduate courses, and for graduate students who wish to learn about quantum cohomology." —Zentralblatt Math

"…The book is ideal for self-study, as a text for a mini-course in quantum cohomology, or 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"   —Analele Stiintifice ale Universitatii “Al. I. Cuza” din Iasi

• Depto. Matematica, Universitat Autònoma de Barcelona Fac. Ciències, Bellaterra, Spain

  Joachim Kock

• Universidade Federal de Pernambuco Depto. Matematica, Recife, Brazil

  Israel Vainsencher