A Real Reduction of the Manifold of Bridgeland Stability Conditions

TL;DR

This paper introduces a quotient space of Bridgeland stability conditions called reduced stability conditions, which simplifies the structure while retaining key features, and explores their properties especially for derived categories of algebraic varieties.

Contribution

It defines a new equivalence relation on stability conditions, constructs the reduced space, and relates it to the original, providing a simpler framework for understanding stability conditions on derived categories.

Findings

Reduced stability space is a real manifold of half the dimension of the original.

The wall-and-chamber structure is preserved in a simpler form.

Existence of stability conditions on a variety implies existence on all its subvarieties.

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 $X$.