DIFFERENTIAL GERSTENHABER–BATALIN–VILKOVISKY ALGEBRAS FOR CALABI–YAU HYPERSURFACE COMPLEMENTS

Abstract

Barannikov and Kontsevich [Frobenius manifolds and formality of Lie algebras of polyvector fields. Int. Math. Res. Not. IMRN 1998(4) (1998), 201215], constructed a DGBV (differential Gerstenhaber-Batalin-ovisky) algebra t for a compact smooth Calabi-Yau complex manifold M of dimension m, which gives rise to the B-side formal Frobenius manifold structure in the homological mirror symmetry conjecture. The cohomology of the DGBV algebra t is isomorphic to the total singular cohomology H-center dot (M) = circle plus(2m)(k=0) H-k (M,C) of M. if M = X-G (C), where X-G is the hypersurface defined by a homogeneous polynomial G((x) under bar) in the projective space P-n, then we give a purely algorithmic construction of a DGBV algebra A(U) , which computes the primitive part circle plus(m)(k=0) PHk of the middle-dimensional cohomology circle plus(m)(k=0) H-k (M,C) using the de Rham cohomology of the hypersurface complement U-G := P-n \\ X-G and the residue isomorphism from H-dR(k) (U-G / C) to PHk. We observe that the DGBV algebra A(U) still makes sense even for a singular projective Calabi-Yau hypersurface, i.e. A(U) computes circle plus(m)(k=0) H-dR(k) (U-G/C) even for a singular X-G. Moreover, we give a precise relationship between A(U) and and t when X-G is smooth in P-n.