  In the first section of this note we show that the Theorem 1.8.1 of
Bayer--Manin ([BaMa]) can be strengthened in the following way: {\it if the
even quantum cohomology of a projective algebraic manifold $V$ is generically
semi--simple, then $V$ has no odd cohomology and is of Hodge--Tate type.} In
particular, this addressess a question in [Ci].
  In the second section, we prove that {\it an analytic (or formal)
supermanifold $M$ with a given supercommutative associative
$\Cal{O}_M$--bilinear multiplication on its tangent sheaf $\Cal{T}_M$ is an
$F$--manifold in the sense of [HeMa], iff 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.
