Joachim's web pages
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.
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
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:
- 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
IMPA, Rio de Janeiro, 1999, xiv + 112pp. (
Colóquio Brasileiro de Matemática, 1999.)
- Tangency quantum cohomology and enumerative geometry of
PhD thesis, Universidade Federal de Pernambuco,
Recife, Brazil, 2000, xx + 155pp. Résumé in:
- Tangency quantum cohomology and characteristic
An. Acad. Bras. Cienc. 73 (2001), 319-326.
- 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.
Last updated: 2009-02-03 by