Communications in Number Theory and Physics

Volume 10 (2016)

Number 2

Enhanced homotopy theory for period integrals of smooth projective hypersurfaces

Pages: 235 – 337



Jae-Suk Park (Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, Gyeongbuk, Korea)

Jeehoon Park (Department of Mathematics, Pohang University of Science and Technology (POSTECH), Pohang, Gyeongbuk, Korea)


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, $L_{\infty}$-homotopy theory).

Let $X_G$ be a smooth projective hypersurface in the complex projective space $\mathbf{P}^n$ defined by a homogeneous polynomial $G(\underline{x})$ of degree $d \geq 1$. Let $\mathbb{H} = H^{n-1}_{\mathrm{prim}} (X_G, \mathbb{C})$ be the middle dimensional primitive cohomology of $X_G$. We explicitly construct a $\mathbf{B \! V \!}$ algebra $\mathbf{B \! V \!}_X = (\mathcal{A}_X, Q_X, K_X)$ such that its $0$-th cohomology $H^0_{K_X} (\mathcal{A}_X)$ is canonically isomorphic to $\mathbb{H}$. We also equip $\mathbf{B \! V \!}_X$ with a decreasing filtration and a bilinear pairing which realize the Hodge filtration and the cup product polarization on $\mathbb{H}$ under the canonical isomorphism. Moreover, we lift $C_{[\gamma]} : \mathbb{H} \to \mathbb{C}$ to a cochain map $\mathcal{C}_{\gamma} : (\mathcal{A}_X, K_X) \to (\mathcal{C}, 0)$, where $ C_{[\gamma]}$ is the Griffiths period integral given by $\omega \mapsto \int_{\gamma} \omega$ for $[\gamma] \in \mathbb{H}_{n-1} (X_G, \mathbb{Z})$.

We use this enhanced homotopy structure on $\mathbb{H}$ 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 $X_{G_{\underline{T}}}$, 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 $X_G$ and an $L_{\infty}$-morphism $\underline{\kappa}$ which enhances $ C_{[\gamma]}$ and governs deformations of period matrices.

2010 Mathematics Subject Classification

13D10, 14D15, 14J70, 18G55

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