Mini-walls for bridgeland stability conditions on the derived category of sheaves over surfaces

Abstract

For the derived category of bounded complexes of sheaves on a smooth projective surface, Bridgeland [Bri2] and Arcara-Bertram [ABL] constructed Bridgeland stability conditions(Zm,Pm) parametrized by m ∈ (0,+∞). In this paper, we show that the set of mini-walls in (0,+Infin;) of a fixed numerical type is locally finite. In addition, we strengthen a result of Bayer [Bay] by proving that the moduli of polynomial Bridgeland semistable objects of a fixed numerical type coincides with the moduli of (Zm,Pm)-semistable objects whenever m is larger than a universal constant depending only on the numerical type. We further identify the moduli of polynomial Bridgeland semistable objects with the Gieseker/Simpson moduli spaces and the Uhlenbeck compactification spaces.