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

Chenjing Bu, Andrés Ibáñez Núñez, Tasuki Kinjo

TL;DR

This work extends intrinsic Donaldson–Thomas theory to general $(-1)$-shifted symplectic stacks by introducing stability measures on the component lattice and constructing epsilon motives as motivic enumerative invariants. It develops a comprehensive motivic framework based on rings of motives, Hall induction, and a Möbius-convolution perspective, culminating in numeric and motivic DT invariants $\mathrm{DT}_\mathcal{X}^{(k)}(\mu)$ and $\mathrm{DT}_\mathcal{X}^{(k),\mathrm{mot}}(\mu)$ defined via $\epsilon_\mathcal{X}^{(k)}(\mu)$ and Behrend-type weightings. A central technical achievement is the no-pole theorem, ensuring the invariants are well-defined after suitable rescaling by powers of $(\mathbb{L}-1)$, with a detailed treatment of Theta-stratifications and their interaction with stability data. The paper lays a robust, intrinsic foundation for wall-crossing and further enumerative structures, with planned applications towall-crossing and multiple-cover phenomena in Part III.

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.

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

TL;DR

This work extends intrinsic Donaldson–Thomas theory to general -shifted symplectic stacks by introducing stability measures on the component lattice and constructing epsilon motives as motivic enumerative invariants. It develops a comprehensive motivic framework based on rings of motives, Hall induction, and a Möbius-convolution perspective, culminating in numeric and motivic DT invariants and defined via and Behrend-type weightings. A central technical achievement is the no-pole theorem, ensuring the invariants are well-defined after suitable rescaling by powers of , with a detailed treatment of Theta-stratifications and their interaction with stability data. The paper lays a robust, intrinsic foundation for wall-crossing and further enumerative structures, with planned applications towall-crossing and multiple-cover phenomena in Part III.

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 -shifted symplectic derived Artin stacks, generalizing the constructions of Joyce-Song and Kontsevich-Soibelman for moduli stacks of objects in -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.

Paper Structure

This paper contains 24 sections, 12 theorems, 128 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The component lattice
  3. Graded and filtered points
  4. The component lattice
  5. The constancy and finiteness theorems
  6. The Hall category and associativity
  7. Linear moduli stacks
  8. Rings of motives
  9. Rings of motives
  10. Schematic realization
  11. Euler characteristics
  12. Hall induction
  13. Stability measures
  14. Stability measures
  15. Möbius convolution for categories
  16. ...and 9 more sections

Key Result

Lemma 5.1.5

The operators $\pi_\mathcal{X}^{(k)}$ are uniquely defined by the properties in para-def-virtual-rank. Moreover, in the integral eq-def-pi-k, we may replace $\mathsf{Face}^\mathrm{sp} (\mathcal{Z})$ by any larger finite full subcategory $\mathcal{F} \subset \mathsf{Face}^\mathrm{nd} (\mathcal{Z})$,

Figures (2)

  • Figure 1: The component lattice of $\mathcal{X}_\gamma$
  • Figure 2: Component lattices for the two quivers

Theorems & Definitions (22)

  • Example 3.4.2: Linear moduli stacks
  • Example 4.1.5: Trivial measure
  • Example 4.1.6: Canonical measure
  • Example 4.1.7: Linear moduli stacks
  • Example 5.1.2: Linear moduli stacks
  • Lemma 5.1.5
  • Theorem 5.1.7
  • Theorem 5.1.8
  • Lemma 5.2.5
  • Example 5.2.6: Virtual rank projections
  • ...and 12 more