Progress in Mathematics

An Invitation to Quantum Cohomology

Kontsevich's Formula for Rational Plane Curves

Authors: Kock, Joachim, Vainsencher, Israel

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About this Textbook

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.

Reviews

"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


Table of contents (7 chapters)

Table of contents (7 chapters)

Buy this book

eBook $64.99
price for USA in USD
  • ISBN 978-0-8176-4495-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • Immediate eBook download after purchase and usable on all devices
  • Bulk discounts available
Hardcover $84.99
price for USA in USD
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Bibliographic Information

Bibliographic Information
Book Title
An Invitation to Quantum Cohomology
Book Subtitle
Kontsevich's Formula for Rational Plane Curves
Authors
Series Title
Progress in Mathematics
Series Volume
249
Copyright
2007
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Boston
eBook ISBN
978-0-8176-4495-6
DOI
10.1007/978-0-8176-4495-6
Hardcover ISBN
978-0-8176-4456-7
Series ISSN
0743-1643
Edition Number
1
Number of Pages
XIV, 162
Topics