A continuity theorem for families of sheaves on complex surfaces

Nicholas Buchdahl (1) , Andrei Teleman (2) , Matei Toma (3)

1 University of Adelaide
2 I2M - Institut de Mathématiques de Marseille
3 IECL - Institut Élie Cartan de Lorraine

We prove that any flat family (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^{ss} := \{u\in 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.

Mots clés: instantons; Donaldson polynomial invariants; Stable vector-bundles; Class-VII surfaces; Moduli spaces; anti-selfdual connections

Domaines: Variables complexes [math.CV]; Géométrie différentielle [math.DG]

Nicholas Buchdahl, Andrei Teleman, Matei Toma. A continuity theorem for families of sheaves on complex surfaces. Journal of topology, 2017, 10 (4), pp.995 - 1028. ⟨10.1112/topo.12029⟩. ⟨hal-01712230⟩