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An Invitation to Quantum Cohomology: Kontsevich's Formula for Rational Plane Curves

Joachim Kock and Israel Vainsencher
Publication Date: 
Number of Pages: 
Progress in Mathematics 249
[Reviewed by
Michael Berg
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The title, by itself, speaks of marvelous and mysterious things: “quantum” somehow never loses its aura of mystery and (if you’ll excuse the quarky puns) strangeness and charm, while cohomology qualifies as arguably the definitive methodological movement of the twentieth century, straddling geometry, topology, and algebra (or any combination of these) and still going strong today.

The juxtaposition of these two themes has of late proven fecund indeed, as is exemplified by the subtitle of the book under review: “Kontsevich’s formula for rational plane curves.” And so we meet one of the major players in this hypermodern game right away: Maxim Kontsevich. The formula the authors feature in this nice book is very beautiful: it gives, in recursive form, the number of rational plane curves of degree d passing through 3d – 1 points in general position. Specifically, we have that this once elusive number is the sum, taken over pairs of lower degrees adding up to d, of a remarkably simple combinatorial expression, making you wonder why, in fact, the formula succeeded in eluding enumerative geometry for so long. But it did, and that just adds to the attraction of the topics presented in Invitation to Quantum Cohomology.

The authors, Joachim Kock and Israel Vainsencher, originally composed the book (in Portuguese) as a companion to a mini-course given in 1999 at IMPA in Rio de Janeiro. They note that Kontsevich originally presented his formula in 1994 as “a corollary” to “the revolution [launched] when a connection between theoretical physics (string theory) and enumerative geometry was discovered.” Kock and Vainsencher then go on to say that Kontsevich ‘s formula “expresses the associativity of a certain new multiplication law, the quantum product” defined on quantum cohomology. This is very exciting and tantalizing stuff!

An Invitation to Quantum Cohomology is the second iteration of the IMPA original, now in English, coming almost a decade later, and has been streamlined for the present larger audience. Kock and Vainsencher note that the reader should meet a few modest prerequisites, namely the (ubiquitous) first chapter of Hartshorne’s Algebraic Geometry and parts of Joseph Harris’ Algebraic Geometry: A First Course. They also stress that theirs is an Invitation only and is properly a prelude to Fulton and Pandharipande’s Notes on Stable Maps and Quantum Cohomology, available gratis on-line.

The book consists of five chapters, taking the reader from moduli spaces, stable curves, and stable maps to Gromov-Witten invariants and quantum cohomology. It is very well-written and accessible, but it is not for raw recruits. However, for those with some familiarity with enumerative geometry, i.e. the aforementioned prerequisites, this is an Invitation that cannot be passed up.

Michael Berg is Professor of Mathematics at Loyola Marymount University.

 Preface.- Introduction.- Prologue: Warming Up with Cross-ratios, and the Definition of Moduli Space.- Stable n-pointed Curves.- Stable Maps.- Enumerative Geometry via Stable Maps.- Gromov–Witten Invariants.- Quantum Cohomology.- Bibliography.- Index.