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

Barannikov and Kontsevich [Frobenius manifolds and formality of Lie algebras of polyvector fields. Int. Math. Res. Not. IMRN 1998 (4) (1998), 201–215], constructed a DGBV (differential Gerstenhaber–Batalin–Vilkovisky) 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 cohomologyH • ( M ) = ⨁ k = 0 2 mH k ( M , C )of M . If M = X G ( C ) , where X G is the hypersurface defined by a homogeneous polynomial G ( x ̲ ) in the projective space P n , then we give a purely algorithmic construction of a DGBV algebra A U , which computes the primitive part⨁ k = 0 m PH kof the middle‐dimensional cohomology⨁ k = 0 m H k ( M , C ) , using the de Rham cohomology of the hypersurface complementU G : = P n ∖ X Gand the residue isomorphism fromH dR k ( U G / C )to PH k . We observe that the DGBV algebra A U still makes sense even for a singular projective Calabi–Yau hypersurface, i.e. A U computes⨁ k = 0 m H dR k ( U G / C )even for a singular X G . Moreover, we give a precise relationship between A U and t when X G is smooth in P n .