A short proof of the deformation property of Bridgeland stability conditions
type: Item
instance of: scholarly article
title: A short proof of the deformation property of Bridgeland stability conditions
zbMATH Open document ID: 1448.14019
author: Arend Bayer
published in: Mathematische Annalen
publication date: 2019-11-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1606.02169
review text: Let \(\mathcal D\) be a triangulated category, and let \(v : K(D) \to \Lambda\) be a homomorphism from its \(K\)-group to a finitely generated free abelian group. Let \(\sigma=(Z,\mathcal P)\) be pre-stability condition on \(\mathcal D\) with respect to \(v\) consisting of a slicing \(\mathcal P\) for \(\mathcal D\) and a centeral charge compatible group homomorphism \(Z : \Lambda \to \mathbb C\). Suppose that \(\sigma\) satisfies the support property with respect to a given quadratic form \(Q:\Lambda_{\mathbb R} \to \mathbb R\) in the sense that \(\ker Z \subset \Lambda_R\) is negative definite with respect to \(Q\), and \(Q(v(E)) \ge 0\) for any semistable object \(E\). In this case, \(\sigma\) is called a stability condition. Let \(\operatorname{Stab}_\Lambda(\mathcal D)\) denote the topological space of stability conditions, and \(\mathcal Z : \operatorname{Stab}_\Lambda(\mathcal D) \to \text{Hom}(\Lambda, \mathbb C)\) be the map \(\mathcal Z(\sigma) = Z\). The main result of the paper under review is a short and direct proof of the following fact: There is an open neighborhood \(U_\sigma\) of \(\sigma\in \operatorname{Stab}_\Lambda(\mathcal D)\) such that the restriction \(\mathcal Z|_{U_\sigma}\) is a covering of the set of \(Z'\in \text{Hom}(\Lambda, \mathbb C)\) such that \(Q\) is negative definite on \(\ker Z'\). Moreover, all stability conditions in \(U_\sigma\) satisfy the support property with respect to \(Q\). This, in particular, proves that \(\operatorname{Stab}_\Lambda(\mathcal D)\) is a manifold as was established by Bridgeland before. The paper also discusses an application of the main result in the case that \(\mathcal D\) is a 2-Calabi-Yau category.
reviewed by: Amin Gholampour
Mathematics Subject Classification ID: 14F08
zbMATH DE Number: 7126540
zbMATH Keywords: triangulated category
zbMATH Keywords: stability condition
MaRDI profile type: Publication
OpenAlex ID: W2972835084
arXiv ID: 1606.02169
cites work: The minimal model program for the Hilbert scheme of points on \(\mathbb P^2\) and Bridgeland stability
cites work: Stability conditions for Slodowy slices and real variations of stability
cites work: Wall-crossing implies Brill-Noether: Applications of stability conditions on surfaces
cites work: Derived automorphism groups of \(K3\) surfaces of Picard rank 1
cites work: The space of stability conditions on the local projective plane
cites work: MMP for moduli of sheaves on \(K3\)s via wall-crossing: nef and movable cones, Lagrangian fibrations
cites work: The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds
cites work: Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities
cites work: Stability conditions on triangulated categories
cites work: Stability conditions on \(K3\) surfaces
cites work: Q3622185
cites work: Stability Conditions and Kleinian Singularities
cites work: Quadratic differentials as stability conditions
cites work: Wall-crossing, Hitchin systems, and the WKB approximation
cites work: Introduction to stability conditions
cites work: Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
cites work: Birational models of moduli spaces of coherent sheaves on the projective plane
cites work: Quotient categories, stability conditions, and birational geometry
cites work: Lectures on Bridgeland Stability
cites work: Degeneration and specialization in algebraic families of vector bundles
cites work: Derived category of coherent sheaves and counting invariants
Wikidata QID: Q127282491
DOI: 10.1007/S00208-019-01900-W
description: scientific article