ON SEMISIMPLE QUANTUM COHOMOLOGY AND F-MANIFOLDS

This answers a question of V. Ginzburg.Finally, we discuss these results in the context of mirror symmetry and Landau–Ginzburg models for Fano varieties.§0. Introduction0.1. Contents of the paper. Semisimple Frobenius manifolds have manynice properties: see e. g. [Du], [Ma], [Te], [Go1], [Go2], and references therein.It is important to understand as precisely as possible, which projective algebraicmanifolds V have (generically) semi–simple quantum cohomology. In this case thequantum cohomology is determined by a finite amount of numbers, and a mirror(Landau–Ginzburg model) can in many cases be described explicitly.If V has non–trivial odd cohomology, its full quantum cohomology cannot besemi–simple, but its even part is a closed Frobenius subspace, and in principle itcan be semisimple. In [BaMa], Theorem 1.8.1, it was proved that if H