Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher rank DT/PT wall-crossing in Bridgeland stability

Marcos Jardim, Jason Lo, Antony Maciocia, Cristian Martinez

TL;DR

The paper demonstrates that the higher-rank DT/PT correspondence on smooth threefolds of Picard rank one is realized as a Bridgeland wall-crossing phenomenon, with a single Θ_v wall separating Gieseker and PT moduli for a fixed Chern character $v$. It develops a comprehensive framework linking PT stability, the large-volume limit, vertical stability, and stable triples, and shows that PT stability corresponds to σ_{∞,B}-stability in the appropriate limit. By analyzing the Θ_v curve and DT/PT wall-crossing, it provides a precise description of how moduli spaces transform across walls, including examples of collapsing, fake, and honest walls. A key consequence is the finiteness of actual walls in the upper half-plane, enabling a robust, arithmetic-geometric understanding of wall-crossing behavior and the moduli of higher-rank PT-stable objects. The results also tie to projectivity results for PT moduli and illuminate how Bridgeland stability captures higher-rank DT/PT phenomena through stable triples and their PT realizations.

Abstract

We prove that the Gieseker moduli space of stable sheaves on a smooth projective threefold $X$ of Picard rank 1 is separated from the moduli space of PT stable objects by a single wall in the space of Bridgeland stability conditions on $X$, thus realizing the higher rank DT/PT correspondence as a wall-crossing phenomenon in the space of Bridgeland stability conditions. In addition, we also show that only finitely many walls pass through the upper $(β,α)$-plane parametrizing geometric Bridgeland stability conditions on $X$ which destabilize Gieseker stable sheaves, PT stable objects or their duals when $α>α_0$.

Higher rank DT/PT wall-crossing in Bridgeland stability

TL;DR

The paper demonstrates that the higher-rank DT/PT correspondence on smooth threefolds of Picard rank one is realized as a Bridgeland wall-crossing phenomenon, with a single Θ_v wall separating Gieseker and PT moduli for a fixed Chern character . It develops a comprehensive framework linking PT stability, the large-volume limit, vertical stability, and stable triples, and shows that PT stability corresponds to σ_{∞,B}-stability in the appropriate limit. By analyzing the Θ_v curve and DT/PT wall-crossing, it provides a precise description of how moduli spaces transform across walls, including examples of collapsing, fake, and honest walls. A key consequence is the finiteness of actual walls in the upper half-plane, enabling a robust, arithmetic-geometric understanding of wall-crossing behavior and the moduli of higher-rank PT-stable objects. The results also tie to projectivity results for PT moduli and illuminate how Bridgeland stability captures higher-rank DT/PT phenomena through stable triples and their PT realizations.

Abstract

We prove that the Gieseker moduli space of stable sheaves on a smooth projective threefold of Picard rank 1 is separated from the moduli space of PT stable objects by a single wall in the space of Bridgeland stability conditions on , thus realizing the higher rank DT/PT correspondence as a wall-crossing phenomenon in the space of Bridgeland stability conditions. In addition, we also show that only finitely many walls pass through the upper -plane parametrizing geometric Bridgeland stability conditions on which destabilize Gieseker stable sheaves, PT stable objects or their duals when .

Paper Structure

This paper contains 25 sections, 42 theorems, 186 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background material
  3. Notation
  4. Stability for sheaves
  5. Technical lemmas
  6. Bridgeland stability conditions on threefolds
  7. Vertical stability
  8. Polynomial stability
  9. Stable triples
  10. Stable triples
  11. Standard representation of triples
  12. Constructing stable triples
  13. PT stability
  14. Properties of PT-semistable objects
  15. Vertical stability and the large volume limit
  16. ...and 10 more sections

Key Result

Proposition 2.2

Let $X$ be a smooth projective variety with an ample class $\omega$, and $\{E_t\}_{t \in I}$ a bounded set of $\mu_\omega$-semistable sheaves on $X$. Then there exists a constant $b \in \mathbb{R}$ such that $\widehat{\mu}_{\mathrm{max}}( (E_t)^{\ast\ast}/(E_t)) < b$ for all $t \in I$.

Figures (3)

  • Figure 1: The picture illustrates the Gieseker and PT chambers in the $(\beta,\alpha)$-plane for a given Chern character $v$. The red curve is the largest tilt wall for $v$, with the point of intersection with the blue $\Theta_v$ curve marked with $(\beta_v,\alpha_v)$, following the notation of Main Theorem \ref{['mthm1']}. The black curve represents the first actual wall that destabilizes Gieseker semistable sheaves and PT semistable objects, as described in Main Theorem \ref{['mthm2']}; this is a piecewise smooth curve, each continuous segment is an arc of a cubic or quartic curve.
  • Figure 2: The four regions of the plane as defined by the hyperbola $\Theta_v$ and the vertical line $\{\beta=\mu(v)\}$ when $v$ is a numerical Chern character satisfying $\Delta(v)\geq 0$.
  • Figure 3: Arrangements of $\rho_i$ for DT and PT polynomial stability conditions on a threefold.

Theorems & Definitions (85)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 75 more