Joachim's web pages [Home] [Math]

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

And here are some notes:
Last updated: 2009-02-03 by Joachim Kock.