Global critical chart for local Calabi-Yau threefolds

TL;DR

This paper explores the derived category of local Calabi-Yau threefolds, describing their structure via deformed Calabi-Yau completions and analyzing the geometry of their moduli stacks as derived critical loci.

Contribution

It provides a new description of the derived category of local Calabi-Yau threefolds using deformed Calabi-Yau completions and characterizes their moduli stacks as derived critical loci.

Findings

Derived category of total space of an $oldsymbol{ ext{ω}_Y}$-torsor is a deformed Calabi-Yau completion.

Derived moduli stack of compactly supported sheaves is equivalent to a derived critical locus.

Application to the study of Higgs bundles and Gopakumar--Vafa invariants.

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, i.e., a Calabi--Yau threefold of the form $\mathrm{Tot}_C(N)$, 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.