A∞-category of a Complex Manifold

Let X be a compact complex manifold equipped with a hermitian metric. In this note we define an A∞-category Db ∞(X) which is a refined version of the derived category of X. This A∞-category is a mirror partner of the A∞-category F (M) constructed by Fukaya for a symplectic manifold M (see [1]). The objects of Db ∞(X) are bounded complexes of holomorphic vector bundles on X equipped with hermitian metrics. The morphisms from E• to F • are elements of Ext(E•, F •) = ⊕qH(X, (E•)∗ ⊗ F •) which can be thought of as harmonic (0, q)-forms with values in (E•)∗ ⊗ F •. The A∞-structure has m1 = 0 whilem2 is the usual composition of Ext’s. In particular, forgetting higher products we obtain the usual derived category of X. The higher products measure in some sense to which extent the product of harmonic forms with values in hermitian bundles fails to be harmonic. More precisely, we use the construction of A∞-structure on a subcomplex homotopy equivalent to a dg-algebra (see [3], [4], [6]). This construction gives an A∞-structure on the algebra Ext(E,E) where E is a hermitian holomorphic vector bundle (or complex of such bundles). It follows from the results of the homological perturbation theory (see [4]) that up to homotopy equivalence our A∞-structures do not depend on choices of metrics. Recall that the homological mirror symmetry conjecture due to M. Kontsevich asserts that some derived category of the Fukaya A∞-category of a Calabi-Yau manifold M (see [5]) is equivalent to the derived category of a mirror dual Calabi-Yau manifold X. We conjecture that moreover in this situation there exists an A∞-functor from F (M) to Db ∞(X) which is a quasi-isomorphism on morphisms. In [9] it was shown that the derived category of an elliptic curve is equivalent to the Fukaya category of special lagrangians in a symplectic torus (disregarding the higher products). In section 2 we show that the corresponding two kinds of (transversal) triple products on the subcategory of line bundles are canonically homotopic to each other.