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 $H^2(X,\mathcal{O}_X) = 0$.

中文翻译：

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对退化的 Calabi-Yau 品种进行平滑处理

我们应用我们之前与 Leung 合作开发的技术来研究一对 $(X,\mathfrak{C}^*)$ 的平滑，其中 $\mathfrak{C}^*$ 是局部自由层的有界完美复形在退化的 Calabi-Yau 品种 $X$ 上。特别地，如果 $X$ 是一个射影 Calabi-Yau 变体，它承认一个环形交叉空间的结构并且具有全局生成的更高切线层 $\mathcal{T}^1_X$，并且 $\mathfrak{F}$ 是一个$X$ 上的局部自由层，然后我们使用 Felten-Filip-Ruddat 的最新结果证明，当 $\text{Ext}^2( \mathfrak{F},\mathfrak{F})_0 = 0$ 和 $H^2(X,\mathcal{O}_X) = 0$。