2025/06/27 by Li, Chunyi
#14F08 #14J60 #14K05 #18G80 #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2506.21995
Let T be a k-linear triangulated category. The space of Bridgeland stability conditions on T, denoted by Stab(T), forms a complex manifold. In this paper, we introduce an equivalence relation ∼ on Stab(T) and study the quotient space Sb(T) := Stab(T)/∼, which parametrizes what we call reduced stability conditions. We show that Sb(T) admits the structure of a real (possibly non-Hausdorff) manifold of half the dimension of Stab(T). The space Sb(T) preserves the wall-and-chamber structure of Stab(T), but in a significantly simpler form. Moreover, we define a relation \lesssim on Sb(T), and show that the full stability manifold Stab(T) can be reconstructed from the space Sb(T) together with the additional data \lesssim. We then focus on the case where T = Db(X), the bounded derived category of coherent sheaves on a smooth polarized variety (X, H). By explicitly describing Sb(X) for varieties X of small dimension, we formulate two equivalent conjectures concerning a family of stability conditions StabH^*(X) and their reduced counterparts SbH^*(X) on Db(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 StabH^*(X) implies the existence of stability conditions on every smooth subvariety of X.