An Update on Semisimple Quantum Cohomology and $F$-Manifolds

In the first section of this note, we show that Theorem 1.8.1 of Bayer–Manin can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold $V$ is generically semisimple, then $V$ has no odd cohomology and is of Hodge–Tate type. In particular, this answers a question discussed by G. Ciolli. In the second section, we prove that an analytic (or formal) supermanifold $M$ with a given supercommutative associative $\mathcal O_M$-bilinear multiplication on its tangent sheaf $\mathcal T_M$ is an $F$-manifold in the sense of Hertling–Manin if and only if its spectral cover, as an analytic subspace of the cotangent bundle $T^*_M,$ is coisotropic of maximal dimension. 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.