[Paper Review] Period integrals of smooth projective hypersurfaces and homotopy Lie algebras

Jae-Suk Park, Jeehoon Park|arXiv (Cornell University)|Oct 24, 2013

Advanced Topics in Algebra3 citations

TL;DR

This paper constructs a Batalin-Vilkovisky (BV) algebra structure on the cohomology of smooth projective hypersurfaces, realizing the primitive cohomology as the zeroth cohomology of this algebra. It lifts Griffiths period integrals to cochain maps and provides an explicit algorithm using Gröbner bases to compute period matrices of Calabi-Yau hypersurface families via an $L_inity$-morphism, revealing hidden homotopy Lie algebra structures underlying period integrals.

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 $\mathbb{H}=H^{n-1}_{\operatorname{prim}}(X_G, \mathbb{C})$ 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 $C_{[\gamma]}:\mathbb{H} o \mathbb{C}$ to a cochain map $\mathcal{C}_\gamma:(\mathcal{A}_X, K_X) o (\mathbb{C},0)$, 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 Grobner 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.

Motivation & Objective

• To uncover hidden BV algebra and $L_inity$-homotopy structures in the singular cohomology of smooth projective hypersurfaces.
• To realize the primitive cohomology $\mathbb{H} = H^{n-1}_{\text{prim}}(X_G, \mathbb{C})$ as the zeroth cohomology of a BV algebra $\mathbf{B\!V}_{\!X}$.
• To equip the BV algebra with a filtration and pairing that reflect the Hodge filtration and cup product polarization on $\mathbb{H}$.
• To lift the Griffiths period integral $C_{[\gamma]}$ to a cochain map $\mathcal{C}_\gamma$ from $\mathcal{A}_X$ to $\mathbb{C}$, enhancing its homotopical structure.
• To apply the enhanced homotopy structure to study formal deformations of Calabi-Yau hypersurfaces and compute period matrices in families.

Proposed method

• Construct a BV algebra $\mathbf{B\!V}_{\!X} = (\mathcal{A}_X, Q_X, K_X)$ such that $H^0_{K_X}(\mathcal{A}_X) \cong \mathbb{H}$, the primitive cohomology.
• Introduce a decreasing filtration on $\mathcal{A}_X$ that induces the Hodge filtration on $\mathbb{H}$ under the canonical isomorphism.
• Define a bilinear pairing on $\mathcal{A}_X$ that realizes the cup product polarization on $\mathbb{H}$.
• Lift the period integral $C_{[\gamma]}: \mathbb{H} \to \mathbb{C}$ to a cochain map $\mathcal{C}_\gamma: (\mathcal{A}_X, K_X) \to (\mathbb{C}, 0)$, encoding higher homotopy data.
• Use the enhanced structure to define an $L_\infty$-morphism $\underline{\kappa}$ that governs deformations of period matrices in formal families of Calabi-Yau hypersurfaces.
• Develop an algorithm based on Gröbner bases to compute the period matrix of $X_{G_{\underline{T}}}$ from the period matrix of $X_G$ and the $L_\infty$-morphism $\underline{\kappa}$.

Experimental results

Research questions

• RQ1 How can BV algebra and $L_\infty$-homotopy structures be realized on the cohomology of smooth projective hypersurfaces?
• RQ2 Can the Griffiths period integral be lifted to a cochain map that encodes higher homotopical data in the cohomology complex?
• RQ3 How do the Hodge filtration and cup product polarization emerge from the algebraic structure of the BV algebra?
• RQ4 What is the role of the $L_\infty$-morphism $\underline{\kappa}$ in governing period matrix deformations in families of Calabi-Yau hypersurfaces?
• RQ5 Can an algorithm be constructed to compute period matrices of deformed Calabi-Yau hypersurfaces using Gröbner basis techniques?

Key findings

• The primitive cohomology $\mathbb{H}$ is canonically isomorphic to the zeroth cohomology $H^0_{K_X}(\mathcal{A}_X)$ of the constructed BV algebra $\mathbf{B\!V}_{\!X}$.
• The BV algebra $\mathbf{B\!V}_{\!X}$ carries a decreasing filtration that induces the Hodge filtration on $\mathbb{H}$ under the canonical isomorphism.
• A bilinear pairing on $\mathcal{A}_X$ realizes the cup product polarization on $\mathbb{H}$, preserving the Hodge-theoretic structure.
• The period integral $C_{[\gamma]}$ is lifted to a cochain map $\mathcal{C}_\gamma$ from $\mathcal{A}_X$ to $\mathbb{C}$, encoding higher homotopy data.
• An explicit algorithm based on Gröbner bases computes the period matrix of a deformed Calabi-Yau hypersurface $X_{G_{\underline{T}}}$ from the period matrix of $X_G$ and the $L_\infty$-morphism $\underline{\kappa}$.
• The $L_\infty$-morphism $\underline{\kappa}$ governs the deformation of period matrices in formal families of Calabi-Yau hypersurfaces, providing a complete homotopical framework for period computations.