Stabilizer Reduction for Derived Stacks and Applications to Sheaf-Theoretic Invariants

TL;DR

This paper introduces a canonical stabilizer reduction for derived algebraic stacks, generalizing classical desingularization techniques to derived geometry, and applies it to define invariants in moduli problems and enumerative geometry.

Contribution

It constructs a derived stabilizer reduction via derived Kirwan blow-ups, extending classical methods to derived stacks with applications to invariants in algebraic geometry.

Findings

Constructed a canonical stabilizer reduction for derived stacks.

Preserves quasi-smoothness and generalizes Kirwan's desingularization.

Defined new invariants for moduli stacks and Calabi-Yau threefolds.

Abstract

We construct a canonical stabilizer reduction $\widetilde{X}$ for any derived $1$-algebraic stack $X$ over $\mathbb{C}$ as a sequence of derived Kirwan blow-ups, under mild natural conditions that include the existence of a good moduli space for the classical truncation $X_{\mathrm{cl}}$. Our construction has several desired features: it naturally generalizes Kirwan's classical partial desingularization algorithm to the context of derived algebraic geometry, preserves quasi-smoothness, and is a derived enhancement of the intrinsic stabilizer reduction constructed by Kiem, Li and the third author. Moreover, if $X$ is $(-1)$-shifted symplectic, we show that the semi-perfect and almost perfect obstruction theory of $\widetilde{X}_{\mathrm{cl}}$ and the associated virtual fundamental cycle and virtual structure sheaf, constructed by the same authors, are naturally induced by $\widetilde{X}$ and its derived tangent complex. As corollaries, we define virtual classes for moduli stacks of semistable sheaves on surfaces, give a fully derived perspective on generalized Donaldson-Thomas invariants of Calabi-Yau threefolds and define new generalized Vafa-Witten invariants for surfaces via Kirwan blow-ups.