(Semi)simple exercises in quantum cohomology

Abstract

The paper is dedicated to the study of algebraic manifolds whose quantum cohomology or a part of it is a semisimple Frobenius manifold. Theorem 1.8.1 says, roughly speaking, that the sum of $(p,p)$--cohomology spaces is a maximal Frobenius submanifold that has chances to be semisimple. Theorem 1.8.3 provides a version of the Reconstruction theorem, assuming semisimplicity but not $H^2$--generation. Theorem 3.6.1 establishes the semisimplicity for all del Pezzo surfaces, providing an evidence for the conjecture that semisimplicity is related to the existence of a full system of exceptional sheaves of the appropriate length. Finally, in \S 2 we calculate special coordinates for three families of Fano threefolds with minimal cohomology.