Jan Segert
University of Missouri

Frobenius Varieties and Quantum Cohomology

Dubrovin's "Frobenius manifolds" are the natural differential-geometric setting for Quantum Cohomology, and for other geometric problems. A Frobenius manifold is a complex manifold with tensor fields that satisfy a certain system of nonlinear partial differential equations. We formulate a global algebro-geometric framework for semisimple Frobenius manifolds of dimension three (and also for the closely-related Painleve VI equation). This allows a large class of Frobenius manifolds to be explicitly constructed as algebraic varieties with rational tensor fields. The construction uses GIT to obtain a projective compactification of a certain moduli space, and Chow's theorem to construct the Frobenius manifolds as subvarieties. The quantum cohomology of the complex projective plane is a particularly interesting example which had heretofore been intractable.