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Gromov-Witten theory

Quantum cohomology and enumerative geometry

Quantum cohomology is concerned with intersection theory on the moduli space of stable maps (from curves to a variety X). The intersection numbers (called Gromov-Witten invariants ) can in nice cases be interpreted as the numbers of curves in X of given genus and degree which satisfy given incidence conditions. One feature is that certain relations among the (genus zero) Gromov-Witten invariants amount to the associativity of the so-called quantum product which is a sort of deformation of the usual cup product in cohomology.

Reference: W. FULTON and R. PANDHARIPANDE. Notes on Stable Maps and Quantum Cohomology.
In: Algebraic Geometry, Santa Cruz 1995, Proc. Symp. Pure. Math. 62 (1997) 45-96. (alg-geom/9608011).

My thesis generalised this to account also for tangency conditions. The highlight is the construction of a tangency quantum product whose associativity provides a nice solution to the characteristic number problem (genus zero). The key notion is a modification of gravitational quantum cohomology which makes some results and methods from theoretical physics available to enumerative geometry.

Reference: You can start for example with this short paper which is a synopsis of my thesis.
Por motivos burocráticos, existe una traducción en español.
Pour des raisons administratives, une version française est disponible.

My publications on Gromov-Witten theory

An invitation to quantum cohomology: Kontsevich's formula for rational plane curves
(with Israel Vainsencher). No. 249 of Progress in Mathematics, Birkhäuser, 2006, xv+175pp.
This is a revised English translation of
A fórmula de Kontsevich para curvas racionais planas
IMPA, Rio de Janeiro, 1999, xiv + 112pp. ( 22^o Colóquio Brasileiro de Matemática, 1999.)
Tangency quantum cohomology and enumerative geometry of rational curves
PhD thesis, Universidade Federal de Pernambuco, Recife, Brazil, 2000, xx + 155pp. Résumé in:
Tangency quantum cohomology and characteristic numbers
An. Acad. Bras. Cienc. 73 (2001), 319-326. [español] [français]
Descendant invariants and characteristic numbers (with Tom Graber and Rahul Pandharipande)
Amer. J. Math. 124 (2002), 611-647.
Characteristic numbers of rational curves with cusp or prescribed triple contact
Math. Scand. 92 (2003), 223-245.
Tangency quantum cohomology
Compositio Math. 140 (2004), 165-178.

And here are some notes:

Notes on psi classes Preliminary version (v. 06/06/2001). Comments are very welcome.
From Frobenius algebras to Frobenius manifolds
Counting bitangents of a smooth plane curves via stable maps Talk at GAeL 1999.
Le nombre de coniques 5-fois tangentes à une courbe plane générique (d'après A. Gathmann) Exposé UNSA, le 6 mars 2002.
(C'est la version pdf d'une collection de transparents détaillés que je n'ai pas utilisée dans l'exposé.)

Last updated: 2009-02-03 by Joachim Kock.