A Real Reduction of the Manifold of Bridgeland Stability Conditions

Abstract

Let $\mathcal{T}$ be a $k$-linear triangulated category. The space of Bridgeland stability conditions on $\mathcal{T}$, denoted by $\mathrm{Stab}(\mathcal{T})$, forms a complex manifold. In this paper, we introduce an equivalence relation $\sim$ on $\mathrm{Stab}(\mathcal{T})$ and study the quotient space $\mathrm{Sb}(\mathcal{T}) := \mathrm{Stab}(\mathcal{T})/\sim$, which parametrizes what we call reduced stability conditions. We show that $\mathrm{Sb}(\mathcal{T})$ admits the structure of a real (possibly non-Hausdorff) manifold of half the dimension of $\mathrm{Stab}(\mathcal{T})$. The space $\mathrm{Sb}(\mathcal{T})$ preserves the wall-and-chamber structure of $\mathrm{Stab}(\mathcal{T})$, but in a significantly simpler form. Moreover, we define a relation $\lesssim$ on $\mathrm{Sb}(\mathcal{T})$, and show that the full stability manifold $\mathrm{Stab}(\mathcal{T})$ can be reconstructed from the space $\mathrm{Sb}(\mathcal{T})$ together with the additional data $\lesssim$. We then focus on the case where $\mathcal{T} = \mathrm{D}^b(X)$, the bounded derived category of coherent sheaves on a smooth polarized variety $(X, H)$. By explicitly describing $\mathrm{Sb}(X)$ for varieties $X$ of small dimension, we formulate two equivalent conjectures concerning a family of stability conditions $\mathrm{Stab}_H^*(X)$ and their reduced counterparts $\mathrm{Sb}_H^*(X)$ on $\mathrm{D}^b(X)$. We establish some desirable properties for both families. In particular, using a version of the restriction theorem formulated in terms of $\lesssim$, we show that the existence of $\mathrm{Stab}_H^*(X)$ implies the existence of stability conditions on every smooth subvariety of $X$.