Topology of moduli spaces of framed sheaves

TL;DR

This paper investigates the topology and geometric structure of moduli spaces of framed torsion-free sheaves on toric surfaces, revealing their decompositions, irreducibility, and homotopy equivalences.

Contribution

It introduces a filtrable Bialynicki-Birula decomposition for these moduli spaces, establishing their irreducibility and homotopy type equivalence to certain subvarieties.

Findings

Moduli spaces admit a filtrable Bialynicki-Birula decomposition.

Irreducibility of the moduli space is established.

Homotopy type matches that of a proper invariant subvariety.

Abstract

In this paper we show that the moduli space of framed torsion-free sheaves on a certain class of toric surfaces admits a filtrable Bialynicki-Birula decomposition determined by the torus action. The irreducibility of this moduli space follows immediately as a corollary. Moreover, using its filtrable decomposition we show that the moduli space shares the same homotopy type of an invariant proper compact subvariety having the same fixed points set. We start our study from the moduli space of framed torsion-free sheaves on the projective plane. Afterwards, we generalize the results to toric surfaces under a few assumptions.