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∞-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.