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# Communications in Number Theory and Physics

## Volume 10 (2016)

### Number 2

### Enhanced homotopy theory for period integrals of smooth projective hypersurfaces

Pages: 235 – 337

DOI: https://dx.doi.org/10.4310/CNTP.2016.v10.n2.a3

#### Authors

#### 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, $L_{\infty}$-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

Published 19 July 2016