Global Critical Chart for Local Calabi–Yau Threefolds

Abstract

Abstract In this paper, we investigate Keller’s deformed Calabi–Yau completion of the derived category of coherent sheaves on a smooth variety. In particular, for an $n$-dimensional smooth variety $Y$, we describe the derived category of the total space of an $\omega _{Y}$-torsor as a certain deformed $(n+1)$-Calabi–Yau completion of the derived category of $Y$. As an application, we investigate the geometry of the derived moduli stack of compactly supported coherent sheaves on a local curve, that is, a Calabi–Yau threefold of the form $\textrm{Tot}_{C}(N)$, where $C$ is a smooth projective curve and $N$ is a rank two vector bundle on $C$. We show that the derived moduli stack is equivalent to the derived critical locus of a function on a certain smooth moduli space. This result will be used by the first author and Naoki Koseki in their joint work on Higgs bundles and Gopakumar–Vafa invariants.