[Paper Review] Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties

Kwokwai Chan, Naichung Conan Leung+1 more|arXiv (Cornell University)|Feb 28, 2019

Geometry and complex manifolds51 references11 citations

TL;DR

This paper constructs an almost differential graded Batalin-Vilkovisky algebra (almost dgBV) from degenerate Calabi-Yau varieties with local smoothing data, providing a unified framework that proves unobstructed smoothing under Hodge-to-de Rham degeneracy and Hodge bundle freeness. It establishes a Bogomolov-Tian-Todorov-type unobstructedness theorem and demonstrates a logarithmic Frobenius manifold structure on the formal extended moduli space.

ABSTRACT

Given a degenerate Calabi-Yau variety $X$ equipped with local deformation data, we construct an almost differential graded Batalin-Vilkovisky (dgBV) algebra $PV^{*,*}(X)$, producing a singular version of the extended Kodaira-Spencer differential graded Lie algebra (dgLa) in the Calabi-Yau setting. Assuming Hodge-to-de Rham degeneracy and a local condition that guarantees freeness of the Hodge bundle, we prove a Bogomolov-Tian-Todorov--type unobstructedness theorem for smoothing of singular Calabi-Yau varieties. In particular, this provides a unified proof for the existence of smoothing of both $d$-semistable log smooth Calabi-Yau varieties (as studied by Friedman and Kawamata-Namikawa and maximally degenerate Calabi-Yau varieties (as studied by Kontsevich-Soibelman and Gross-Siebert). We also demonstrate how our construction yields a logarithmic Frobenius manifold structure on a formal neighborhood of $X$ in the extended moduli space by applying the technique of Barannikov-Kontsevich.

Motivation & Objective

• To develop a singular analogue of the Kodaira-Spencer dgBV algebra for degenerate Calabi-Yau varieties.
• To unify the proofs of smoothing existence for both d-semistable log smooth and maximally degenerate Calabi-Yau varieties.
• To establish a Bogomolov-Tian-Todorov-type unobstructedness theorem for singular Calabi-Yau smoothing under Hodge-to-de Rham degeneracy and Hodge bundle freeness.
• To construct a logarithmic Frobenius manifold structure on the formal extended moduli space near a maximally degenerate Calabi-Yau variety.

Proposed method

• Constructs an almost dgBV algebra $\prescript{}{}{PV}^{*,*}(X)$ via a local-to-global Čech-de Rham gluing procedure from local deformation data on a degenerate Calabi-Yau variety $X$.
• Applies a simplicial construction to capture nontrivial topology change due to non-locally trivial thickenings in the deformation process.
• Uses divisorial log deformation theory inspired by Gross-Siebert to define local models $\mathbb{V}_{\tau}^\dagger$ and their thickenings $\prescript{k}{}{\mathbb{V}}^\dagger_{\tau}$.
• Employs the Barannikov-Kontsevich technique to derive a logarithmic Frobenius manifold structure from the extended moduli space of $X$.
• Relies on Hodge-to-de Rham degeneracy and a local freeness condition on the Hodge bundle to ensure unobstructedness of deformations.
• Utilizes a weight filtration opposite to the Hodge filtration and defines a trace map to construct a pairing $\prescript{0}{}{\mathtt{p}}$ for Frobenius structure.

Experimental results

Research questions

• RQ1 Can a singular analogue of the Kodaira-Spencer dgBV algebra be constructed to govern smoothing of degenerate Calabi-Yau varieties?
• RQ2 Does the proposed almost dgBV algebra control geometric smoothings through its Maurer-Cartan equation?
• RQ3 Can a unified unobstructedness theorem for smoothing be derived for both d-semistable and maximally degenerate Calabi-Yau varieties?
• RQ4 Does the extended moduli space of a maximally degenerate Calabi-Yau variety admit a logarithmic Frobenius manifold structure?
• RQ5 Under what conditions is the pairing $\prescript{0}{}{\mathtt{p}}$ non-degenerate, enabling the Frobenius structure?

Key findings

• An almost dgBV algebra $\prescript{}{}{PV}^{*,*}(X)$ is constructed from local deformation data on a degenerate Calabi-Yau variety $X$, capturing nontrivial topology change via non-locally trivial thickenings.
• Under Hodge-to-de Rham degeneracy and Hodge bundle freeness, the Maurer-Cartan equation of $\prescript{}{}{PV}^{*,*}(X)$ governs unobstructed smoothings of $X$.
• The construction provides a unified proof of smoothing existence for both $d$-semistable log smooth and maximally degenerate Calabi-Yau varieties.
• A logarithmic Frobenius manifold structure is shown to exist on the formal extended moduli space $\hat{S}_{T}^\dagger$ near $X$, assuming the pairing $\prescript{0}{}{\mathtt{p}}$ is non-degenerate.
• The cohomological decomposition of differential forms on $\mathsf{V}_{\tau} \setminus \mathscr{Z}$ is expressed in terms of $P_{\tau}$-homogeneous pieces and intersections of divisor groups.
• The trace map $\operatorname{tr}$ induces an isomorphism $H^d(X, j_*(\Omega^d_{X^\dagger/\prescript{0}{}{S}^\dagger})) \cong \mathbb{C}$, supporting the duality needed for Frobenius structure.