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On the Donaldson-Uhlenbeck compactification of instanton moduli spaces on class VII surfaces. (English) Zbl 1442.14108

Let \(X\) be a compact class \(VII\) surface equipped with a Gauduchon metric \(g\) and denote by \(\mathcal M^{\text{ASD}}(E)\) the moduli space of \(\operatorname{SU}_2\) instantons on \(M\) with \(c_2 = 1\). Let also denote by \(\mathcal M^{\text{ASD}}(E)^*\), \(\mathcal R\) the subspaces of \(\mathcal M^{\text{ASD}}(E)\) made of the irreducible and reducible instantons and by \(\mathcal M_0^*\), \(\mathcal R_0\) the moduli spaces of flat irreducible and flat reducible \(\operatorname{SU}_2\)-instantons, respectively.
It is known that:
(1) By the Kobayashi-Hitchin correspondence, \( \mathcal M^{\text{ASD}}(E)^*\) is homeomorphic to the moduli space \( \mathcal M^{\text{st}}(E)^*\) of stable holomorphic structures on \(E\) and is thus equipped with a natural complex structure.
(2) Donaldson’s compactification \(\overline{\mathcal M^{\text{ASD}}(E)}\) of \(M^{\text{ASD}}(E)\) is equal to \[\overline{\mathcal M^{\text{ASD}}(E)} = \mathcal M^{\text{ASD}}(E)^* \cup \mathcal R \cup (\mathcal M_0^* \times X) \cup (\mathcal R_0 \times X)\ .\] The sets \(\mathcal R\), \(\mathcal M_0^* \times X\), \(\mathcal R_0 \times X\) are known to be compact
In this paper the authors focus on the following question: Does the natural complex space structure of \( \mathcal M^{\text{ASD}}(E)^*\) extend across the compact strata \(\mathcal R\), \( \mathcal M_0^* \times X\) and \(\mathcal R_0 \times X\)? The motivation for this problem comes from the interest for the conditions under which moduli spaces of irreducible stable vector bundles on compact Gauduchon manifolds can be considered as open subsets of appropriate complex compactifications.
The authors consider in detail the extendibility of the complex structure of \(\mathcal M^{\text{ASD}}(E)^* \) across each of the above three strata. They find that there exists a natural smooth extension of the complex space structure across the points of \(\mathcal R_0 \times X\) and a (possibly non-smooth) extension across the points of \(\mathcal M_0^* \times X\). The picture is completed by the results in v, which imply that the complex space structure is in general not extendible across \(\mathcal R\) when \(b_2 >0\). However, using the previous results and sharp Teleman’s analysis the structure of \(\overline{\mathcal M^{\text{ASD}}(E)}\) around the points \(\mathcal M_0^* \times X\) under the condition \( \text{deg}_g(\mathcal K_X) < 0\), the authors finally obtain the following
Corollary. Let \(X\) be a primary Hopf surface equipped with a Gauduchon metric \(g\) and \(E\) an \(\operatorname{SL}_2(\mathbb C)\)-bundle on \(X\) with \(c_1 = 1\). Then the natural complex structure of \( \mathcal M^{\text{st}}(E)^*\) is smooth and extends to a complex structure on \(\overline{\mathcal M^{\text{ASD}}(E)}\), which becomes in this way a \(4\)-dimensional compact complex manifold.

MSC:

14H60 Vector bundles on curves and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
32Q26 Notions of stability for complex manifolds
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