Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Compact self-dual Hermitian surfaces


Authors: Vestislav Apostolov, Johann Davidov and Oleg Muskarov
Journal: Trans. Amer. Math. Soc. 348 (1996), 3051-3063
MSC (1991): Primary 53C55
DOI: https://doi.org/10.1090/S0002-9947-96-01585-1
MathSciNet review: 1348147
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we obtain a classification (up to conformal equivalence) of the compact self-dual Hermitian surfaces. As an application, we prove that every compact Hermitian surface of pointwise constant holomorphic sectional curvature with respect to either the Riemannian or the Hermitian connection is Kähler.


References [Enhancements On Off] (What's this?)

  • [1] M.F.Atiyah, N.J.Hitchin, I.M.Singer, Self-duality in four-dimensional Riemannian geometry, Proc.Roy.Soc. London Ser.A 362 (1978), 425-461. MR 80d:53023
  • [2] A.Balas, Compact Hermitian Manifolds of Constant Holomorphic Sectional Curvature, Math.Z. 189 (1985), 193-210. MR 86f:53072
  • [3] A.Balash, P.Gauduchon, Any Hermitian Metric of Constant Non-Positive (Hermitian) Holomorphic Sectional Curvature is Kähler, Math.Z. 190 (1985), 39-43. MR 86h:53066
  • [4] A.Besse, Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete 3.Folge, Band 10, Springer, Berlin Heildelberg New-York, 1987. MR 88f:53087
  • [5] J.P.Bourguignon, Les variétés de dimension 4 à signature non nule dont la courbure est harmonique sont d'Einstein, Invent. Math. 63 (1981), 263-286. MR 82g:53051
  • [6] Ch.Boyer, Conformal duality and compact complex surfaces, Math. Ann. 274 (1986), 517-526. MR 87i:53068
  • [7] Ch.Boyer, Self-dual and anti-self-dual Hermitian metrics on compact complex surfaces, Mathematics and general relativity, Proceedings, Santa Cruz 1986 (J.A.Isenberg, ed.), Contemp.Math., vol. 71, AMS, Providence, 1988, pp. 105-114. MR 89h:53127
  • [8] B.-Y.Chen, Some topological obstructions to Bochner-Kähler metrics and their applications, J.Diff.Geom. 13 (1978), 574-588. MR 81f:32037
  • [9] A.Derdzinski, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), 405-433. MR 84h:53060
  • [10] Th.Friedrich, H.Kurke, Compact four-dimensional self-dual Einstein manifolds with positive scalar curvature, Math.Nachr. 106 (1982), 271-299. MR 84b:53043
  • [11] P.Gauduchon, Le théorème de l'excentricité nulle, C.R.Acad.Sci.Paris, Ser.A 285 (1977), 387-390. MR 57:10664
  • [12] P.Gauduchon, Fibrés Hermitiens à endomorphisme de Ricci non négatif, Bull.Soc.Math. France 105 (1977), 113-140. MR 58:6375
  • [13] P.Gauduchon, Self-dual manifolds with non-negative Ricci operator, Prospects in complex geometry, Proceedings, Katata and Kyoto 1989 (J.Noguchi, T.Ohsawa, eds.), Lecture Notes in Math., vol. 1468, Springer, Berlin Heidelberg New York, 1991, pp. 55-61. MR 94g:53032
  • [14] G.Grantcharov, O.Mu\v{s}karov, Hermitian *-Einstein surfaces, Proc.Amer.Math.Soc. 120 (1994), 233-239. MR 94b:53081
  • [15] N.J.Hitchin, Compact four-dimensional Einstein manifolds,J.Diff.Geom. 9 (1974), 435-441. MR 50:3149
  • [16] N.J.Hitchin, Kählerian twistor spaces, Proc.London Math.Soc. 43 (1981), 133-150. MR 84b:32014
  • [17] M.Itoh, Self-duality of Kähler surfaces, Compositio Math. 51 (1984), 265-273. MR 85m:53079
  • [18] T.Koda, Self-dual and anti-self-dual Hermitian surfaces, Kodai Math. J. 10 (1987), 335-342. MR 89a:53053
  • [19] T.Koda, K.Sekigawa, Self-dual Einstein Hermitian surfaces, Progress in Diff.Geom. (K.Shiohama, ed.), Advanced Studies in Pure Mathematics, vol.22, Kinokuniya, Tokyo, 1993, pp. 123-131. MR 95b:53056
  • [20] K.Kodaira, On the structure of compact complex analytic surfaces, II, Amer.J.Math. 88 (1968), 682-721. MR 34:5112
  • [21] J.Lafontaine, Remarks sur les variété conformément plates, Math.Ann. 259 (1982), 313-319. MR 84a:53053
  • [22] Y.Miyaoka, On Chern numbers of surfaces of general type, Invent.Math. 42 (1977), 225-237. MR 57:337
  • [23] M.H.Noronha, Self-duality and 4-manifolds with nonnegative curvature on totally isotropic 2-planes, Michigan Math.J. 41(1994), 3-12. MR 95e:53069
  • [24] M.Pontecorvo, Uniformization of conformally flat Hermitian surfaces, Diff.Geom. and its Appl. 2 (1992), 295-305. MR 94k:32052
  • [25] Y.Poon, Compact self-dual manifolds with positive scalar curvature, J.Diff.Geom. 24 (1986), 97-132. MR 88b:32022
  • [26] T.Sato, K.Sekigawa, Hermitian surfaces of constant holomorphic sectional curvature, Math.Z. 205 (1990), 659-668. MR 91m:53052
  • [27] R.Schoen, Conformal deformations of Riemannian metrics to constant scalar curvature, J.Diff.Geom. 20 (1984), 479-495. MR 86i:58137
  • [28] K.Sekigawa, T.Koda, Compact Hermitian surfaces of pointwise constant holomorphic sectional curvature, to appear in Glasgow Math.J.
  • [29] I.M.Singer, J.A.Thorpe, The curvature of 4-dimensional Einstein spaces, Global analysis, Papers in honor of K.Kodaira (D.C.Spencer, S.Iyanaka, eds.), Princeton University Press, Princeton, 1969, pp. 355-365. MR 41:959
  • [30] F.Tricerri, L.Vanhecke, Curvature tensors on almost-Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), 365-398. MR 82j:53071
  • [31] I.Vaisman, Some curvature properties of complex surfaces, Ann.Mat.Pura Appl. 32 (1982), 1-18. MR 84i:53064
  • [32] I.Vaisman, Generalized Hopf manifolds, Geom. Dedicata 13 (1982), 231-255. MR 84g:53096
  • [33] W.T.Wu, Sur la structure presque complex d'une variété différentiable réelle de dimension 4, C.R.Acad.Sci.Paris 227 (1948), 1076-1078. MR 10:318b

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 53C55

Retrieve articles in all journals with MSC (1991): 53C55


Additional Information

Vestislav Apostolov
Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. bl.8, 1113 Sofia, Bulgaria

Johann Davidov
Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. bl.8, 1113 Sofia, Bulgaria
Email: jtd@bgearn.bitnet

Oleg Muskarov
Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. bl.8, 1113 Sofia, Bulgaria

DOI: https://doi.org/10.1090/S0002-9947-96-01585-1
Received by editor(s): December 13, 1994
Additional Notes: Research parially supported by the Bulgarian Ministry of Science and Education, contract MM-423/94.
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society