Derived category of coherent systems on curves and stability conditions

TL;DR

The paper introduces a stability-theoretic framework for the derived category of coherent systems on a smooth projective curve $C$ of genus $g>0$, showing that an open two-dimensional slice of Bridgeland stability conditions on $\mathcal{D}(\mathcal{T}_C)$ is governed by two complementary constructions: tilting stability along $\mu$-stability enriched by Brill-Noether data and gluing stability along a natural semi-orthogonal decomposition. It proves that, up to the $\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$-action, any stability condition in the open locus $\mathrm{Stab}^{\circ}(\mathcal{D}(\mathcal{T}_C))$ is either a gluing $\operatorname{gl}^{(1)}(\sigma_{\mathcal{V}},\sigma_g)$ with $f(0)<\tfrac{1}{2}$ or a tilting type described by $(b,w)$ with $w>\Phi_C(b)$, where $\Phi_C$ is the Brill-Noether function of $C$. The main results connect wall-crossing in this two-dimensional slice to the Brill-Noether theory of $C$, relate large-volume behavior to classical $\alpha$-stability for coherent systems, and, in a second gluing, classify stability conditions arising from the semi-orthogonal decompositions, linking geometric stability of vector bundles to the derived category of coherent systems. These constructions yield a precise description of the stability manifold $\mathrm{Stab}^{\circ}(\mathcal{D}(\mathcal{T}_C))$ as a union of explicit open sets, and provide a robust bridge between Bridgeland stability, coherent systems, and Brill-Noether theory with potential applications to moduli and birational geometry of vector bundles on curves.

Abstract

Let $C$ be a smooth projective curve of genus $g>0$. We describe an open locus of Bridgeland stability conditions on the bounded derived category of coherent systems on $C$, and show that stability manifold detects the Brill--Noether theory of the curve.

Theorems & Definitions (75)

• Theorem 1.1: = Theorem \ref{['thm-main']}
• Corollary 1.2: = Corollary \ref{['cor complex manifold']}
• Theorem 1.3: = Theorem \ref{['thm-second open subset']}
• Lemma 2.1
• proof
• Proposition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof