Degenerations of shifted symplectic moduli stacks of sheaves

Abstract

Let $S$ be a smooth, affine noetherian $\mathbb{C}$-scheme, and let $X/S$ be a smooth, quasi-projective family of Calabi--Yau $d$-folds. We prove that the derived moduli stack of flat families of properly supported coherent sheaves on $X/S$ has a relative $(2-d)$-shifted symplectic structure by generalizing a construction of Brav--Dyckerhoff. The main points are to show that the category $\mathrm{IndCoh}(X)$ is a smooth, compact $\mathrm{QCoh}(S)$-module and to identify our moduli stack of interest as a substack of the moduli of pseudo-perfect objects in $\mathrm{IndCoh}(X)$. Finally, we discuss two examples of families of smooth, quasi-projective Calabi--Yau threefolds that are potentially useful for applications in enumerative geometry.