Unobstructed deformations for singular Calabi-Yau varieties

TL;DR

This work extends unobstructed deformation results for singular Calabi–Yau varieties beyond weighted-homogeneous and Kähler assumptions by using a resolution $\hat Y$ satisfying the $\partial\bar{\partial}$-lemma and the condition $H^1(Y;\mathcal{O}_Y)=0$, for isolated Du Bois singularities. It proves that if the singularities are also local complete intersections, the deformations of $Y$ are unobstructed, and it generalizes the $T^1$ lifting framework to the non-lci setting via a generalized lifting property. The paper develops a Hodge-theoretic machinery for relative cohomology to control deformations, including $E_1$-degeneration for relative cohomology and surjectivity results for deformation cohomology $H^{n-2}(\mathcal{Y};\Omega^1_{\mathcal{Y}/\operatorname{Spec}A})$, which underpin unobstructedness. It further extends unobstructedness to log Calabi–Yau pairs $(Y,D)$ and to Fano-type scenarios, employing cyclic covers and log deformation theory to link pair and space deformations. Overall, the results place Du Bois singularities in a natural position for unobstructed deformation theory of singular Calabi–Yau varieties and connect with prior birational and deformation frameworks (Kawamata–Ran–Tian, Namikawa, Gross, Imagi).

Abstract

Let $Y$ be a compact Gorenstein analytic space with only isolated singularities and trivial dualizing sheaf. A recent paper of Imagi studies the deformation theory of $Y$ in case the singularities of $Y$ are weighted homogeneous and rational and $Y$ is Kähler. In this note, assuming that $H^1(Y;\mathcal{O}_Y) =0$, we generalize Imagi's results to the case where the singularities of $Y$ are Du Bois, with no assumption that they be weighted homogeneous, and where the Kähler assumption is replaced by the hypothesis that there is a resolution of singularities of $Y$ satisfying the $\partial\bar\partial$-lemma. As a consequence, if the singularites of $Y$ are additionally local complete intersections, then the deformations of $Y$ are unobstructed. The log Calabi-Yau and Fano cases are also discussed.

Theorems & Definitions (52)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 2.1
• proof
• Lemma 2.2