An Invitation to Quantum Cohomology: Kontsevich's Formula for Rational Plane Curves
An Invitation to Quantum Cohomology: Kontsevich's Formula for Rational Plane Curves (Progress in Mathematics) Joachim Kock, Israel Vainsencher,
Publisher: Birkhäuser Boston
ISBN: 0817644563
DDC: 516.35
Edition: Hardcover; 2006-10-24
Summary:
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 GromovWitten 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.
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