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

Given a degenerate Calabi–Yau variety $X$ equipped with local deformation data, we construct an almost differential graded Batalin–Vilkovisky algebra $PV^{\ast,\ast}(X)$, producing a singular version of the extended Kodaira–Spencer differential graded Lie algebra 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 [$\href{https://doi.org/10.2307/2006955}{22}$] and Kawamata–Namikawa [$\href{ https://doi.org/10.1007/BF01231538}{41}$]) and maximally degenerate Calabi–Yau varieties (as studied by Kontsevich–Soibelman $[\href{ https://link.springer.com/chapter/10.1007/0-8176-4467-9_9}{45}$] and Gross–Siebert [$\href{ http://doi.org/10.4007/annals.2011.174.3.1}{30}$]). 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 [$\href{https://doi.org/10.1155/S1073792898000166}{2}$, $\href{https://doi.org/10.48550/arXiv.math/9903124}{1}$].