Perverse coherent sheaves on blow-up. II. wall-crossing and Betti numbers formula

TL;DR

This paper investigates the moduli spaces of stable perverse coherent sheaves on the blow-up of a surface, describing wall-crossings, Betti number formulas, and relations to classical moduli spaces.

Contribution

It provides a detailed analysis of wall-crossing phenomena and Betti number formulas for these moduli spaces, linking them to classical sheaf moduli spaces on original and blown-up surfaces.

Findings

Wall-crossing between moduli spaces caused by twisting line bundles.

Explicit formula for virtual Hodge numbers of the moduli spaces.

Isomorphisms between moduli spaces under certain conditions.

Abstract

This is the second of series of papers studyig moduli spaces of a certain class of coherent sheaves, which we call stable perverse coherent sheaves, on the blow-up of a projective surface at a point. The followings are main results of this paper: a) We describe the wall-crossing between moduli spaces caused by twisting of the line bundle associated with the exceptional divisor. b) We give the formula for virtual Hodge numbers of moduli spaces of stable perverse coherent sheaves. Moreover we also give proofs of the followings which we observed in a special case in arXiv:0802.3120: c) The moduli space of stable perverse coherent sheaves is isomorphic to the usual moduli space of stable coherent sheaves on the original surface if the first Chern class is orthogonal to the exceptional divisor. d) The moduli space becomes isomorphic to the usual moduli space of stable coherent sheaves on the blow-up after twisting by sufficiently large negative power of the line bundle associated with the exceptional curve. Therefore usual moduli spaces of stable sheaves on the blow-up and the original surfaces are connected via wall-crossings.