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Enhanced homotopy theory for period integrals of smooth projective hypersurfaces
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- Title
- Enhanced homotopy theory for period integrals of smooth projective hypersurfaces

- Authors
- PARK, JAE SUK; PARK, JEEHOON

- Date Issued
- 2016-06

- Publisher
- International Press

- Abstract
- The goal of this paper is to reveal hidden structures on the singular cohomology and the Griffiths period integral of a smooth projective hypersurface in terms of BV(Batalin-Vilkovisky) algebras and homotopy Lie theory (so called, L-infinity-homotopy theory). Let X-G be a smooth projective hypersurface in the complex projective space P-n defined by a homogeneous polynomial G((x) under bar) of degree d >= 1. Let H = H-prim(n-1) (X-G, C) be the middle dimensional primitive cohomology of X-G(prim). We explicitly construct a BV algebra BVX = (A(X), Q(X), K-X) such that its 0-th cohomology H-KX(0) (A(X)) is canonically isomorphic to H. We also equip BVX with a decreasing filtration and a bilinear pairing which realize the Hodge filtration and the cup product polarization on H under the canonical isomorphism. Moreover, we lift GET] : IRE C to a cochain map l(r) : (A(X), K-X) -> (C, 0), where C-[gamma] is the Griffiths period integral given by omega -> integral(gamma) omega for [gamma] is an element of Hn-1 (X-G, Z). We use this enhanced homotopy structure on H to study an extended formal deformation of X-G and the correlation of its period integrals. If X-G is in a formal family of Calabi-Yau hypersurfaces X-G(T) under bar, we provide an explicit formula and algorithm (based on a Grobner basis) to compute the period matrix of X-G (T) under bar, in terms of the period matrix of X-G (X) under bar and an L-infinity-morphism (kappa) under bar which enhances C-[gamma] and governs deformations of period matrices.

- ISSN
- 1931-4523

- Article Type
- Article

- Citation
- Communications in Number Theory and Physics, vol. 10, no. 2, page. 235 - 337, 2016-06

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- 박지훈PARK, JEEHOON
- Dept of Mathematics

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