A continuity theorem for families of sheaves on complex surfaces

Abstract

We prove that any flat family $({\mathcal{F}}_u{)}_{u\in U}$ of rank 2 torsion-free sheaves on a Gauduchon surface defines a continuous map on the semi-stable locus $U^{\mathrm{ss}}:=\{u\in U|{\mathcal{F}}_{u}is\; slope\; semi-stable\}$ 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 $U^{\mathrm{ss}}$ 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.

References in corpus (1)

• Compact moduli spaces for slope-semistable sheaves

Cited by in corpus (8)

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• Bounded sets of sheaves on relative analytic spaces
• Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation
• On the Donaldson-Uhlenbeck compactification of instanton moduli spaces on class VII surfaces
• Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds
• Moduli spaces of slope-semistable sheaves with reflexive Seshadri graduations