Homological mirror symmetry with higher products

Details Homological mirror symmetry with higher products Homological mirror symmetry with higher products

1999-01-06

arxiv.org/abs/math/9901025v5

Mathematics

Algebraic Geometry

AMSLatex, 13 pages, the final version

Scientific paper

We construct an $A_{\infty}$-structure on the Ext-groups of hermitian holomorphic vector bundles on a compact complex manifold. We propose a generalization of the homological mirror conjecture due to Kontsevich. Namely, we conjecture that for mirror dual Calabi-Yau manifolds $M$ and $X$ there exists an $A_{\infty}$-functor from Fukaya's symplectic $A_{\infty}$-category of $M$ to the $A_{\infty}$-derived category of $X$ which is a homotopy equivalence on morphisms. We verify the part of this conjecture concering triple products for elliptic curves.

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