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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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