2013/10/24 by Park, Jae-Suk, Park, Jeehoon
#13D10 #14D15 #14J70 #18G55 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1310.6710
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(\underline x) of degree d ≥ 1. Let ℍ=Hn-1prim(XG, ℂ) be the middle dimensional primitive cohomology of XG. We explicitly construct a BV algebra B V X=(AX,QX, KX) such that its 0-th cohomology H0KX(AX) is canonically isomorphic to ℍ. We also equip B V X 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↦ ∫γω for [γ]∈ 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 X_G\underline T, we provide an explicit formula and algorithm (based on a Gröbner basis) to compute the period matrix of X_G\underline T in terms of the period matrix of XG and an L_∞-morphism \underline κ which enhances C[γ] and governs deformations of period matrices.