0 Semi - infinite A - variations of Hodge structure over extended Kähler cone

According to Kontsevich's Homological Mirror Conjecture [K1], a mirror pair, X a ˆ X, of Calabi-Yau manifolds has two associated A ∞-categories, the derived category of coherent sheaves on X and the Fukaya category ofˆX, equivalent. In particular, the moduli spaces of A ∞-deformations of these two categories must be isomorphic implying Another expected corollary is the equivalence of two Frobenius manifold structures, the first one is generated on H * (X, ∧ * T X) by the periods of semi-infinite variations of Hodge structure on X [B2, B3], and the second one is generated on H * (ˆ X, C) by the Gromov-Witten invariants. The l.h.s. in the above equality can be identified with the tangent space at X to the extended moduli space, M compl , of complex structures [BK]. This moduli space is the base of semi-infinite B-variations, VHS B (X), of the standard Hodge structure in H * (X, C) [B2]. Moreover, it was shown in [B2] how to construct a family of Frobenius manifold structures, {Φ W compl (X)}, on M compl parameterized by isotropic increasing filtraions, W , in the de Rham cohomology H * (X, C) which are complementary to the standard decreasing Hodge filtration. Presumably, the compactification M compl contains a point with maximal unipotent monodromy, and the associated limiting weight filtration W 0 gives rise, via Barannikov's semi-infinite variations of Hodge structure, to the solution, Φ W 0 compl (X), of the WDVV equations which coincides precisely with the potential, Φ GW (ˆ X), built out of the Gromov-Witten invariants on the mirror side. This has been checked for complete Calabi-Yau intersections in [B1]. It is widely believed that Φ GW (ˆ X) can itself be reconstructed A-model variations of Hodge structure (see [CF, CK, Mo] for the small quantum cohomology case). In this paper we propose a symplectic version of the Barannikov's construction which, presumably, extends the results of [CF, CK, Mo] to the full quantum cohomology group. We study semi-infinite A-variations, VHS A (ˆ X), of Hodge structure over the extended moduli space, M sympl , of Kähler forms on the mirror partnerˆX, and then use Barannikov's technique [B2] to build out of VHS A (ˆ X) a family of solutions of the WDVV equations, {Φ