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ЖУРНАЛЫ // Труды Математического института имени В. А. Стеклова // Архив

Труды МИАН, 2009, том 264, страницы 69–76 (Mi tm804)

Эта публикация цитируется в 24 статьях

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

C. Hertlinga, Yu. I. Maninbc, C. Telemande

a Institut für Mathematik, Universität Mannheim, Mannheim, Germany
b Max-Planck-Institut für Mathematik, Bonn, Germany
c Northwestern University, Evanston, USA
d University of Edinburgh, UK
e University of California, Berkeley, USA

Аннотация: 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.

УДК: 514.743.2

Поступило в июле 2008 г.

Язык публикации: английский


 Англоязычная версия: Proceedings of the Steklov Institute of Mathematics, 2009, 264, 62–69

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