Boyer, Charles P. Conformal duality and compact complex surfaces. (English) Zbl 0571.32017 Math. Ann. 274, 517-526 (1986). 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. Cited in 3 ReviewsCited in 28 Documents MSC: 32J15 Compact complex surfaces 53C55 Global differential geometry of Hermitian and Kählerian manifolds 14J15 Moduli, classification: analytic theory; relations with modular forms Keywords:anti-self-dual metric; Kähler metric; classification of compact complex surfaces × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [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 · doi:10.1098/rspa.1978.0143 [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 · doi:10.1007/BF01393878 [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 · doi:10.1016/0001-8708(74)90021-8 [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 · doi:10.2307/2373157 [17] Lafontaine, J.: Remarques sur les varietes conforment plates. Math. Ann.259, 313-319 (1982) · doi:10.1007/BF01456943 [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 · doi:10.1007/BF01647970 [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) · doi:10.1016/0550-3213(84)90340-7 [22] Vaisman, I.: Some curvature properties of complex surfaces, Ann. Mat. Para Appl.32, 1-18 (1982) · Zbl 0512.53058 · doi:10.1007/BF01760974 [23] Vaisman, I.: Generalized Hopf manifolds. Geom. Dedicata13, 231-255 (1982) · Zbl 0506.53032 · doi:10.1007/BF00148231 [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 · doi:10.1007/BF01389728 [27] Yau, S.T. (Ed.) Seminar on differential geometry. Princeton Princeton Univ. Press 1982 · Zbl 0471.00020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.