Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties
Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties
复制标题
简并 Calabi-Yau 变体附近的 Maurer-Cartan 方程的几何
DOI：
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发表时间：
2019
影响因子：
2.5
通讯作者：
Z. Ma
中科院分区：
文献类型：
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作者：
Kwokwai Chan;N. Leung;Z. Ma
给定配备局部变形数据的简并 Calabi-Yau 变体 X，我们构造了一个几乎微分分级的 Batalin-Vilkovisky（几乎 dgBV）代数 PV(X)，给出了 Calabi-Yau 设置中扩展 Kodaira-Spencer dgLa 的奇异版本。假设 Hodge-to-de Rham 简并性和保证 Hodge 丛自由性的局部条件，我们证明了奇异 Calabi-Yau 簇平滑的 Bogomolov-Tian-Todorov 型无阻碍定理。特别是，这为对数平滑 Calabi-Yau 变体（由 Friedman 和 Kawamata-Namikawa 研究）和最大退化 Calabi-Yau 变体（由 Kontsevich-Soibelman 和 Gross-Siebert 研究）的平滑的存在提供了统一的证明。我们还演示了我们的构造如何使用 Barannikov-Kontsevich 的技术在扩展模空间中的 X 形式邻域上产生对数 Frobenius 流形结构。
Given a degenerate Calabi-Yau variety X equipped with local deformation data, we construct an almost differential graded Batalin-Vilkovisky (almost dgBV) algebra PV(X), giving a singular version of the extended Kodaira-Spencer 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 the smoothing of singular Calabi-Yau varieties. In particular, this provides a unified proof for the existence of smoothing of both log smooth Calabi-Yau varieties (studied by Friedman and Kawamata-Namikawa) and maximally degenerate Calabi-Yau varieties (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 using the technique of Barannikov-Kontsevich.