Intrinsic Donaldson-Thomas theory. II. Stability measures and invariants

TL;DR

This paper develops a new framework for defining Donaldson-Thomas invariants for complex algebraic stacks, generalizing previous constructions and introducing stability measures to ensure well-defined invariants.

Contribution

It constructs intrinsic Donaldson-Thomas invariants for $(-1)$-shifted symplectic derived stacks using motives and stability measures, extending prior work to more general stacks.

Findings

Generalized Joyce's no-pole theorem for stacks

Defined invariants using rings of motives

Established foundational properties for future applications

Abstract

This is the second paper in a series on intrinsic Donaldson-Thomas theory, a framework for studying the enumerative geometry of general algebraic stacks. In this paper, we present the construction of Donaldson-Thomas invariants for general $(-1)$-shifted symplectic derived Artin stacks, generalizing the constructions of Joyce-Song and Kontsevich-Soibelman for moduli stacks of objects in $3$-Calabi-Yau abelian categories. Our invariants are defined using rings of motives, and depend intrinsically on the stack, together with a set of combinatorial data similar to a stability condition, called a stability measure on the component lattice of the stack. For our invariants to be well-defined, we prove a generalization of Joyce's no-pole theorem to general stacks, using a simpler and more conceptual argument than the original proof in the abelian category case. Further properties and applications of these invariants, such as wall-crossing formulae, will be discussed in a forthcoming paper.