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 $X$.
Categories: math.AG math-ph math.MP