Homological mirror symmetry with higher products

Abstract

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.

AMSLatex, 13 pages, the final version