An Update on Semisimple Quantum Cohomology and F-Manifolds

作者：Hertling Claus^*; Manin Yuri I; Teleman Constantin

来源：Proceedings of the Steklov Institute of Mathematics, 2009, 264(1): 62-69.

DOI：10.1134/S0081543809010088

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 O(M)-bilinear multiplication on its tangent sheaf 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.

• 出版日期2009-4
• 单位西北大学