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

다국어 초록 (Multilingual Abstract)

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 cohomology H•(M)=⨁k=02mHk(M,C) of M. If M=XG(C), where XG is the hypersurface defined by a homogeneous polynomial G(x̲) in the projective space Pn, then we give a purely algorithmic construction of a DGBV algebra AU, which computes the primitive part ⨁k=0mPHk of the middle‐dimensional cohomology ⨁k=0mHk(M,C), using the de Rham cohomology of the hypersurface complement UG:=Pn∖XG and the residue isomorphism from HdRk(UG/C) to PHk. We observe that the DGBV algebra AU still makes sense even for a singular projective Calabi–Yau hypersurface, i.e. AU computes ⨁k=0mHdRk(UG/C) even for a singular XG. Moreover, we give a precise relationship between AU and t when XG is smooth in Pn.