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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
Author(s)
Jae-Suk Park; Park J.
Publication Date
2016-06
Journal
COMMUNICATIONS IN NUMBER THEORY AND PHYSICS, v.10, no.2, pp.235 - 337
Publisher
INT PRESS BOSTON
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∞-homotopy theory). Let XG be a smooth projective hypersurface in the complex projective space Pn defined by a homogeneous polynomial G(x) of degree d ≥ 1. Let H = Hn-1 prim(XG, C) be the middle dimensional primitive cohomology of XG. We explicitly construct a BV algebra BVX = (AX,QX,KX) such that its 0-th cohomology H0 K X (AX) 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 C[γ]: H → C to a cochain map Cγ: (AX, KX) → (C, 0), where C[γ] is the Griffiths period integral given by ω → ∫γ ω for [γ] ε Hn-1(XG, Z). We use this enhanced homotopy structure on H to study an extended formal deformation of XG and the correlation of its period integrals. If XG is in a formal family of Calabi-Yau hypersurfaces XGT, we provide an explicit formula and algorithm (based on a Gröbner basis) to compute the period matrix of XGT in terms of the period matrix of XG and an L∞-morphism K which enhances C[γ] and governs deformations of period matrices.
URI
https://pr.ibs.re.kr/handle/8788114/2832
ISSN
1931-4523
Appears in Collections:
Center for Geometry and Physics(기하학 수리물리 연구단) > Journal Papers (저널논문)
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