A continuity theorem for families of sheaves on complex surfaces

Abstract

We prove that any flat family ( F u ) u ∈ U of rank 2 torsion-free sheaves on a Gauduchon surface defines a continuous map on the semi-stable locus U ss : = { u ∈ U | F u is slope semi-stable } with values in the Donaldson–Uhlenbeck compactification of the corresponding instanton moduli space. In the general (possibly non-Kählerian) case, the Donaldson–Uhlenbeck compactification is not a complex space, and the set U ss can be a complicated subset of the base space U that is neither open or closed in the classical topology, nor locally closed in the Zariski topology. This result provides an efficient tool for the explicit description of Donaldson–Uhlenbeck compactifications on arbitrary Gauduchon surfaces.