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Donaldson theory on non-Kählerian surfaces and class VII surfaces with $$b_2=1$$. (English) Zbl 1093.32006
In this very substantial paper it is proved that every compact complex surface $$X$$ of class VII (Kodaira dimension kod$$(X)=-\infty$$ and $$b_1(X)=1$$) contains a compact complex curve. This result finishes the classification problem for surfaces $$X$$ of class VII with $$b_2(X)\leq 1$$. If $$b_2(X)=0$$ then $$X$$ is biholomorphically equivalent to a Hopf surface or to an Inoue surface by a theorem of F. A. Bogomolov [Math. USSR, Izv. 10 (1976), 255–269 (1977), translation from Izv. Akad. Nauk SSSR, Ser. Mat. 40, 273–288 (1976; Zbl 0352.32020)], for a complete proof see also J. Li, S.T, Yau and F. Zheng [Commun. Anal. Geom. 2, No. 1, 103–109 (1994; Zbl 0837.53053)] and A. Teleman [Int. J. Math. 5, No. 2, 253–264 (1994; Zbl 0803.53038)]. The classification in the situation $$b_2(X)=1$$ and $$X$$ containing a curve was given in 1984 by I. Nakamura [Invent. Math. 78, 393–443 (1984; Zbl 0575.14033)].
In the present paper it is shown that if $$b_2(X)=1$$, then there exists an effective divisor $$C>0$$ on $$X$$ such that $$c_1^{\mathbb Q}({\mathcal O}(C))$$ equals one of the rational cohomology classes $$\pm c_1^{\mathbb Q}({\mathcal K}),\,\,0,\,\, 2c_1^{\mathbb Q}({\mathcal K})$$, where $${\mathcal K}$$ denotes the canonical line bundle of $$X$$. The idea of the proof is to show that otherwise there would exist a certain moduli space of stable rank 2 bundles on $$X$$ which contains a compact Riemann surface $$Y$$ with points corresponding to non-filtrable bundles (bundles with no subsheaves of rank 1), and points corresponding to filtrable bundles. This leads at the end to a non-trivial holomorphic map from $$Y$$ into some moduli space and to a contradiction to the fact that the algebraic dimension $$a(X)=0$$. The proof uses techniques from Donaldson theory, compactness theorems for moduli spaces of stable bundles and the Kobayashi-Hitchin correspondence on surfaces.

##### MSC:
 32J15 Compact complex surfaces 14J15 Moduli, classification: analytic theory; relations with modular forms 57R57 Applications of global analysis to structures on manifolds
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##### References:
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