Smoothing Pairs Over Degenerate Calabi–Yau Varieties

Abstract

We apply the techniques developed in [2] 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 results in [ 8], that the pair $(X,\mathfrak{F})$ is formally smoothable when $\textrm{Ext}^2(\mathfrak{F},\mathfrak{F})_0 = 0$ and $H^2(X,\mathcal{O}_X) = 0$.

The logarithmic Bogomolov–Tian–Todorov theorem

“…If $$Q = 0$$, we adapt the formalism of differential graded Lie algebras, which was used by Iacono and Manetti to give an algebraic proof of the BTT theorem [16–18], to the context of log schemes. If $$Q \ne 0$$, we use the recent formalism developed by Chan–Leung–Ma [5] (see also [4]) and by Felten–Filip–Ruddat [8] to construct smoothings of degenerate Calabi–Yau varieties, and we apply an algebraic result (Proposition 2.8).…”

The logarithmic Bogomolov-Tian-Todorov theorem

“…If Q = 0, we adapt the formalism of differential graded Lie algebras, which was used by Iacono and Manetti to give an algebraic proof of the BTT theorem [14][15][16], to the context of log schemes. If Q ≠ 0, we use the recent formalism developed by Chan-Leung-Ma [4] (see also [5]) and by Felten-Filip-Ruddat [8] to construct smoothings of degenerate Calabi-Yau varieties, and we apply an algebraic result (Proposition 2.8).…”