DIFFERENTIAL GERSTENHABER-BATALIN-VILKOVISKY ALGEBRAS FOR CALABI-YAU HYPERSURFACE COMPLEMENTS
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- Title
- DIFFERENTIAL GERSTENHABER-BATALIN-VILKOVISKY ALGEBRAS FOR CALABI-YAU HYPERSURFACE COMPLEMENTS
- Authors
- KIM, DOKYOUNG; KIM, YESEUL; Park, Jeehoon
- Date Issued
- 2018-06
- Publisher
- LONDON MATH SOC
- 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.
- URI
- https://oasis.postech.ac.kr/handle/2014.oak/94630
- DOI
- 10.1112/S0025579318000177
- ISSN
- 0025-5793
- Article Type
- Article
- Citation
- MATHEMATIKA, vol. 64, no. 3, page. 637 - 651, 2018-06
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