Instantons and curves on class VII surfaces. (English) Zbl 1231.14028

A class VII surface is a compact non-Kähler surface with first Betti number \(b_1=1\) and Kodaira dimension \(-\infty\). Class VII surfaces with second Betti number \(b_2=0\) are known to be either a Hopf surface or an Inoue surface. On the other hand, for class VII surfaces with \(b_2>0\) it is not known whether they contain a curve. Results which prove the existence of curves on such surfaces seem to be extremely difficult to obtain.
Thanks to work of M. Kato [Proc.Japan Acad.53, 15–16 (1977; Zbl 0379.32023)] class VII surfaces which contain a global spherical shell (an open surface biholomorphic to the standard neighbourhood of \(S^3\) in \({\mathbb C}^2\)) are well understood. In particular it is known that such surfaces contain a cycle of rational curves. A conjecture of Nakamura states that any minimal class VII surface with \(b_2>0\) contains a global spherical shell. This was shown to be true for \(b_2=1\) in a previous article by the author [Invent.Math.162, No.3, 493–521 (2005; Zbl 1093.32006)]. On the other hand, I. Nakamura [Tohoku Math.J., II.Ser.42, No.4, 475–516 (1990; Zbl 0732.14019)] has shown that a minimal class VII surface which contains a cycle of curves is a specialisation of blown-up primary Hopf surfaces.
The main result of the paper under review states that each minimal class VII surface with \(b_2=2\) contains a cycle of curves. The main tool used in this article is the moduli space of polystable (with respect to a suitable Gauduchon metric) rank-2 vector bundles with \(c_2=0\) and determinant isomorphic to the canonical line bundle. Assuming that a minimal class VII surface with \(b_2=2\) does not contain a cycle of curves, he proves with gauge theoretical techniques that this moduli space is a compact topological manifold and that the subspace corresponding to stable bundles is a complex manifold having a smooth compact connected component which contains only a finite but non-empty subset representing filtrable bundles (bundles which possess subsheaves of rank 1). He then shows that the existence of such a component leads to a contradiction.
The main obstacle to applying the same ideas to the case \(b_3\geq 3\) is that the geometry of the moduli space of a polystable bundle will become very complicated.


14J15 Moduli, classification: analytic theory; relations with modular forms
32J15 Compact complex surfaces
14J25 Special surfaces
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
32G13 Complex-analytic moduli problems
32L05 Holomorphic bundles and generalizations
32Q57 Classification theorems for complex manifolds
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
57R57 Applications of global analysis to structures on manifolds
Full Text: DOI arXiv


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