An update on semisimple quantum cohomology and F-manifolds

TL;DR

This paper strengthens a theorem relating semi-simple quantum cohomology to the Hodge--Tate property, characterizes supermanifolds with F-manifold structures via spectral covers, and discusses implications for mirror symmetry and Landau--Ginzburg models.

Contribution

It improves understanding of semi-simple quantum cohomology and F-manifolds, providing new criteria and connections relevant to mirror symmetry and algebraic geometry.

Findings

Semi-simple quantum cohomology implies no odd cohomology and Hodge--Tate type.

Supermanifolds are F-manifolds iff their spectral cover is coisotropic of maximal dimension.

Results have implications for mirror symmetry and Landau--Ginzburg models.

Abstract

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.