层的平移辛模叠的退化
Degenerations of shifted symplectic moduli stacks of sheaves
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中文总结 AI 辅助
本文通过推广 Brav--Dyckerhoff 的构造，证明了 Calabi--Yau 族上凝聚层的导出模叠具有相对平移辛结构，并识别其为伪完美对象模的子叠，为枚举几何提供了潜在应用。
中文摘要 AI 辅助
设 $S$ 是一个光滑、仿射诺特 $\nmathbb{C}$-概形，且 $X/S$ 是 Calabi--Yau $d$ 维流形的光滑、拟射影族。我们通过推广 Brav--Dyckerhoff 的构造，证明了 $X/S$ 上适当支撑的凝聚层的平坦族的导出模叠具有相对 $(2-d)$-平移辛结构。主要要点是证明范畴 $\mathrm{IndCoh}(X)$ 是一个光滑、紧致的 $\mathrm{QCoh}(S)$-模，并将我们感兴趣的模叠识别为 $\mathrm{IndCoh}(X)$ 中伪完美对象的模的子叠。最后，我们讨论了两个光滑、拟射影 Calabi--Yau 三维流形的族例子，这些例子可能对枚举几何中的应用有用。
英文摘要
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.
