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An update on semisimple quantum cohomology and F-manifolds

  • Claus HertlingEmail author
  • Yuri I. Manin
  • Constantin Teleman
Open Access
Article

Abstract

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.

Keywords

STEKLOV Institute Tangent Sheaf Poisson Structure Cotangent Bundle Hodge Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  • Claus Hertling
    • 1
    Email author
  • Yuri I. Manin
    • 2
    • 3
  • Constantin Teleman
    • 4
    • 5
  1. 1.Institut für MathematikUniversität MannheimMannheimGermany
  2. 2.Northwestern UniversityEvanstonUSA
  3. 3.Max-Planck-Institut für MathematikBonnGermany
  4. 4.University of EdinburghEdinburghUK
  5. 5.University of CaliforniaBerkeleyUSA

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