# Jan SegertUniversity 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.