  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.
