Enhanced homotopy theory for period integrals of smooth projective hypersurfaces

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 $\mathbb{H}=H^{n-1}_{\operatorname{prim}}(X_G, \mathbb{C})$ be the middle dimensional primitive cohomology of $X_G$. We explicitly construct a BV 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 $\mathcal{C}_\gamma:(\mathcal{A}_X, K_X) \to (\mathbb{C},0)$, where $C_{[\gamma]}$ is the Griffiths period integral given by $\omega \mapsto \int_\gamma \omega$ for $[\gamma]\in H_{n-1}(X_G,\mathbb{Z})$. 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 Gr\"obner 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.