Degenerations of shifted symplectic moduli stacks of sheaves

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