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arXiv · 1902.11174

Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties

Kwokwai Chan·Naichung Conan Leung·Ziming Nikolas Ma
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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.

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Kwokwai Chan, Naichung Conan Leung, Ziming Nikolas Ma. 2019-02-28. Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties. https://doi.org/10.4310/jdg/1695236591

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