Enhanced homotopy theory for period integrals of smooth projective hypersurfaces

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.