  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.
