Smoothing pairs over degenerate Calabi-Yau varieties

Abstract

We apply the techniques developed in our previous work with Leung to study smoothings of a pair $(X,{\mathfrak{C}}^{\ast })$, where ${\mathfrak{C}}^{\ast }$ 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 ${\mathcal{T}}_X^1$ 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 $H^2(X,{\mathcal{O}}_X)=0$.

23 pages; v2: final version to appear in IMRN

References in corpus (5)

• Smoothing toroidal crossing spaces
• Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties
• Tropical Lagrangian multi-sections and smoothing of locally free sheaves over degenerate Calabi-Yau surfaces
• Differential Graded Lie Algebras and Deformations of Holomorphic Maps
• On the deformation theory of pair (X, E)

Cited by in corpus (3)

• Smoothing toroidal crossing spaces
• Tropical Lagrangian multi-sections and smoothing of locally free sheaves over degenerate Calabi-Yau surfaces
• Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program