On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator

Details On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator

1994-03-12

arxiv.org/abs/alg-geom/9403010v1

Mathematics

Algebraic Geometry

21 pages, latex with amstex-fonts

Scientific paper

We observe a general structure theorem for quantum cohomology rings, a non-homogeneous version of the usual cohomology ring encoding information about (almost holomorphic) rational curves. An application is the rigorous computation of the quantum cohomology of Grassmannians. As purely algebraic consequence we prove a beautiful formula of Vafa and Intriligator for intersection numbers of certain compactifications of moduli spaces of maps from a Riemann surface (any genus) to G(k,n) which recently has excited many mathematicians. The formula generalizes to any Fano manifold whose cohomology ring can be presented as complete intersection.

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