2021/07/28 by Kotaro Kawatani, Kawatani, Kotaro
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2107.13367
openalex publication_date 2021/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
On a triangulated category \mathbf D equipped with a semiorthogonal decomposition \mathbf D=⟨\mathbf D1,\mathbf D2⟩, Collins and Polishchuk develop a gluing construction of stability condition on \mathbf D. The gluing construction gives a stability condition on \mathbf D from these on \mathbf D1 and \mathbf D2. We study a deformation of gluing stability conditions on for a nice semiorthogonal decomposition. As a consequence, we construct a continuous family of stability conditions by showing a deformation property introduced by Bridgeland's original paper. Here the deformation property is weaker than the support property which is the standard solution for the continuousness. After proving the continuousness of the family, we show that each stability condition in the family satisfies the support property via specialization. More precisely we find a stability condition with support property at the boundary of the family. Finally applying these results, we study the space of stability conditions on the category of morphisms in a triangulated category.