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Conformal duality and compact complex surfaces. (English) Zbl 0571.32017
Let M be a compact complex surface with first Betti number \(b_ 1\) which admits a compatible anti-self-dual metric g. In this paper we prove that if \(b_ 1\) is even, then g is conformally equivalent to a Kähler metric with zero scalar curvature and if \(b_ 1\) is odd then g is conformally equivalent to a locally conformally Kähler metric with positive scalar curvature. Applying this result to the Enriques-Kodaira classification of compact complex surfaces, we give a list of possible surfaces which admit an anti-self-dual metric.

MSC:
32J15 Compact complex surfaces
53C55 Global differential geometry of Hermitian and Kählerian manifolds
14J15 Moduli, classification: analytic theory; relations with modular forms
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[1] Atiyah, M.F.: Geometry of Yang-Mills fields. Academia Nazionale dei Lincei, Scuola Normale Superiore, Pisa (1979) · Zbl 0435.58001
[2] Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. Lond. A362, 425-461 (1978) · Zbl 0389.53011
[3] Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. New York: Springer 1984 · Zbl 0718.14023
[4] Bourguigon, J.P.: Les varietés de dimension 4 a signature non nulle dont la courbure est harmonique sont d’Einstein. Invent. Math.63, 263-286 (1981) · Zbl 0456.53033
[5] Derdzinski, A.: Hermitian Einstein metrics. In: Global Riemannian geometry. Ed. Willmore, T.J., Hitchin, N.J. pp. 105-114
[6] Derdzinski, A.: Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compos. Math.49, 405-433 (1983) · Zbl 0527.53030
[7] Gauduchon, P.: Fibrés Hermitiens a endomorphisme de Ricci non negatif. Bull. Soc. Math. France105, 113-140 (1970) · Zbl 0382.53045
[8] Gauduchon, P.: Le théorème de l’excentricité nulle. C.R. Acad. Sci. Paris285, 387-390 (1977) · Zbl 0362.53024
[9] Goldberg, S.I.: Curvature and homology. New York: Academic Press 1962 · Zbl 0105.15601
[10] Gromov, M., Lawson, H.B.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math. Inst. Haustes Etud. Sci.58, 295-408 (1983) · Zbl 0538.53047
[11] Hitchin, N.: Compact Four-dimensional Einstein Manifolds. J. Differ. Geom.9, 435-441 (1974) · Zbl 0281.53039
[12] Hitchin, N.: Harmonic spinors. Adv. Math.14, 1-55 (1974) · Zbl 0284.58016
[13] Ibrahim, J., Lichnerowicz, A.: Tenseurs holomorphes sur une variéte kahlérienne compacte. C.R. Acad. Sci. Paris,277, 801-805 (1973) · Zbl 0283.53047
[14] Itoh, M.: Self-duality of Kähler surfaces. Compos. Math.51, 265-273 (1984) · Zbl 0546.53044
[15] Kazdan, J.L., Warner, F.W.: Prescribing curvature. Proc. Symp. Pure Math27, 309-319 (1973)
[16] Kodaira, K.: On the structure of compact complex analytic surfaces. I. Am. J. Math.86, 751-798 (1964) · Zbl 0137.17501
[17] Lafontaine, J.: Remarques sur les varietes conforment plates. Math. Ann.259, 313-319 (1982) · Zbl 0478.53034
[18] Lichnerowicz, A.: Spineurs harmoniques. C.R. Acad. Sci. Paris, sér, A-B257, 7-9 (1963)
[19] Schoen, R., Yau, S.T.: On the structure of manifolds with positive scalar curvature, Manuscr. Math.28, 159-183 (1979) · Zbl 0423.53032
[20] Singer, I.M., Thorpe, J.A.: The curvature of 4-dimensional Einstein spaces. In: Global analysis, papers in honor of K. Kodaira. Eds. Spencer, D.C., Iyanaga, S., pp. 355-365 · Zbl 0199.25401
[21] Strominger, A., Horowitz, G.T., Perry, M.J.: Instantons in conformal gravity. Nucl. Phys.B 238, 635-664 (1984)
[22] Vaisman, I.: Some curvature properties of complex surfaces, Ann. Mat. Para Appl.32, 1-18 (1982) · Zbl 0512.53058
[23] Vaisman, I.: Generalized Hopf manifolds. Geom. Dedicata13, 231-255 (1982) · Zbl 0506.53032
[24] Vaisman, I.: On locally and globally conformal Kähler manifolds. Trans. Am. Math. Soc.262, 533-542 (1980) · Zbl 0446.53048
[25] Yano, K., Bochner, S.: Curvature and Betti numbers. Princeton: Princeton Univ. Press 1953 · Zbl 0051.39402
[26] Yau, S.T.: On the curvature of compact Hermitian manifolds. Invent. Math.25, 213-239 (1974) · Zbl 0299.53039
[27] Yau, S.T. (Ed.) Seminar on differential geometry. Princeton Princeton Univ. Press 1982 · Zbl 0471.00020
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