Smoothing pairs over degenerate Calabi-Yau varieties

TL;DR

This paper investigates conditions under which pairs involving degenerate Calabi-Yau varieties can be smoothed, utilizing advanced techniques and recent results to establish formal smoothability criteria.

Contribution

It extends smoothing theory to pairs over degenerate Calabi-Yau varieties, providing new criteria for formal smoothability using complex geometric and cohomological conditions.

Findings

Pairs are formally smoothable under specific Ext and cohomology vanishing conditions.

Utilizes recent results to connect geometric structures with deformation theory.

Applies to projective Calabi-Yau varieties with toroidal crossing structures.

Abstract

We apply the techniques developed in our previous work with Leung to study smoothings of a pair $(X,\mathfrak{C}^*)$, where $\mathfrak{C}^*$ is a bounded perfect complex of locally free sheaves over a degenerate Calabi-Yau variety $X$. In particular, if $X$ is a projective Calabi-Yau variety admitting the structure of a toroidal crossing space and with the higher tangent sheaf $\mathcal{T}^1_X$ globally generated, and $\mathfrak{F}$ is a locally free sheaf over $X$, then we prove, using the recent results of Felten-Filip-Ruddat, that the pair $(X,\mathfrak{F})$ is formally smoothable when $\text{Ext}^2(\mathfrak{F},\mathfrak{F})_0 = 0$ and $H^2(X,\mathcal{O}_X) = 0$.