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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.

32J15 Compact complex surfaces
14J15 Moduli, classification: analytic theory; relations with modular forms
57R57 Applications of global analysis to structures on manifolds
Full Text: DOI arXiv
[1] Barlet, D.: Majoration du volume des fibres génériques et formes géométriques du théorème d’aplatissement. Séminaire Pierre Lelong-Henri Skoda (Analyse). Lect. Notes Math., vol. 822, pp. 1–17. Springer 1980
[2] Barlet, D.: How to use the cycle cpace in complex geometry. Several Complex Variables MSRI Publications, vol. 37, pp. 25–42, 1999 · Zbl 0962.32016
[3] Bănică, C., Le Potier, J.: Sur l’existence des fibrés vectoriels holomorphes sur les surfaces non-algébriques. J. Reine Angew. Math. 378, 1–31 (1987) · Zbl 0624.32017 · doi:10.1515/crll.1987.378.1
[4] Barth, W., Hulek, K., Peters, Ch., Van de Ven, A.: Compact complex surfaces. Springer 2004 · Zbl 1036.14016
[5] Bogomolov, F.: Classification of surfaces of class VII0 with b2=0. Math. USSR Izv. 10, 255–269 (1976) · Zbl 0352.32020 · doi:10.1070/IM1976v010n02ABEH001688
[6] Bogomolov, F.: Surfaces of class VII0 and affine geometry. Math. USSR Izv. 21, 31–73 (1983) · Zbl 0527.14029 · doi:10.1070/IM1983v021n01ABEH001640
[7] Buchdahl, N.: Hermitian-Einstein connections and stable vector bundles over compact complex surfaces. Math. Ann. 280, 625–648 (1988) · Zbl 0617.32044 · doi:10.1007/BF01450081
[8] Buchdahl, N.: A Nakai-Moishezon criterion for non-Kahler surfaces. Ann. Inst. Fourier 50, 1533–1538 (2000) · Zbl 0964.32014
[9] Dloussky, G., Oeljeklaus, K., Toma, M.: Class VII0 surfaces with b2 curves. Tohoku Math. J., II. Ser. 55, 283–309 (2003) · Zbl 1034.32012 · doi:10.2748/tmj/1113246942
[10] Donaldson, S. K.: Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. 50, 1–26 (1985) · Zbl 0547.53019 · doi:10.1112/plms/s3-50.1.1
[11] Donaldson, S., Kronheimer, P.: The Geometry of Four-Manifolds. Oxford Univ. Press 1990 · Zbl 0820.57002
[12] Gauduchon, P.: Sur la 1-forme de torsion d’une variété hermitienne compacte. Math. Ann. 267, 495–518 (1984) · Zbl 0536.53066 · doi:10.1007/BF01455968
[13] Kobayashi, S.: Differential geometry of complex vector bundles. Princeton Univ. Press 1987 · Zbl 0708.53002
[14] Lübke, M., Okonek, C.: Moduli spaces of simple bundles and Hermitian-Einstein connections. Math. Ann. 276, 663–674 (1987) · Zbl 0609.53009 · doi:10.1007/BF01456994
[15] Lübke, M., Teleman, A.: The Kobayashi-Hitchin correspondence. World Scientific Publishing Co. 1995 · Zbl 0849.32020
[16] Lübke, M., Teleman, A.: The universal Kobayashi-Hitchin correspondence on Hermitian surfaces. math.DG/0402341, to appear in Mem. Am. Math. Soc.
[17] Li, J., Yau, S.T.: Hermitian Yang-Mills connections on non-Kähler manifolds, Math. aspects of string theory (San Diego, CA 1986). Adv. Ser. Math. Phys. 1, pp. 560–573. World Scientific Publishing 1987
[18] Li, J., Yau, S.T., Zheng, F.: On projectively flat Hermitian manifolds. Commun. Anal. Geom. 2, 103–109 (1994) · Zbl 0837.53053
[19] Miyajima, K.: Kuranishi families of vector bundles and algebraic description of the moduli space of Einstein-Hermitian connections. Publ. Res. Inst. Math. Sci. 25, 301–320 (1989) · Zbl 0683.32016 · doi:10.2977/prims/1195173613
[20] Nakamura, I.: On surfaces of class VII0 with curves. Invent. Math. 78, 393–443 (1984) · Zbl 0575.14033 · doi:10.1007/BF01388444
[21] Newstead, P.E.: Introduction to moduli problems and orbit spaces. Tata Institute of Fundamental Research on Mathematics and Physics 51. New Delhi: Tata Institute of Fundamental Research 1978 · Zbl 0411.14003
[22] Teleman, A.: Projectively flat surfaces and Bogomolov’s theorem on class VII0-surfaces. Int. J. Math. 5, 253–264 (1994) · Zbl 0803.53038 · doi:10.1142/S0129167X94000152
[23] Teleman, A.: Instantons on class VII surfaces. In preparation · Zbl 1231.14028
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