Asian Journal of Mathematics

Volume 18 (2014)

Number 2

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

Pages: 321 – 344

DOI: http://dx.doi.org/10.4310/AJM.2014.v18.n2.a7

Authors

Jason Lo (Department of Mathematics, University of Missouri, Columbia, Mo., U.S.A.)

Zhenbo Qin (Department of Mathematics, University of Missouri, Columbia, Mo., U.S.A.)

Abstract

For the derived category of bounded complexes of sheaves on a smooth projective surface, Bridgeland and Arcara-Bertram constructed Bridgeland stability conditions $(Z_m , \mathcal{P}_m)$ parametrized by $m \in (0, {+\infty})$. In this paper, we show that the set of mini-walls in $(0, {+\infty})$ of a fixed numerical type is locally finite. In addition, we strengthen a result of Bayer by proving that the moduli of polynomial Bridgeland semistable objects of a fixed numerical type coincides with the moduli of $(Z_m , \mathcal{P}_m)$-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.

Keywords

walls, Bridgeland stability, polynomial stability, derived category

2010 Mathematics Subject Classification

Primary 14D20. Secondary 14F05, 14J60.

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