A short proof of the deformation property of Bridgeland stability conditions
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.
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