Weakly-Einstein hermitian surfaces
Annales de l'Institut Fourier, Volume 49 (1999) no. 5, pp. 1673-1692.

A consequence of the Riemannian Goldberg-Sachs theorem is the fact that the Kähler form of an Einstein Hermitian surface is an eigenform of the curvature operator. Referring to this property as *-Einstein condition we obtain a complete classification of the compact locally homogeneous *-Einstein Hermitian surfaces. We also provide large families of non-homogeneous *-Einstein (but non-Einstein) Hermitian metrics on 2 ¯ 2 , 1 × 1 , and on the product surface X×Y of two curves X and Y whose genuses are greater than 1 and 0, respectively.

Le théorème de Goldberg-Sachs riemannien a pour conséquence le fait que toute surface complexe, hermitienne, d’Einstein satisfait la condition *-Einstein disant que la forme de Kähler est forme propre de l’opérateur de courbure. Dans cet article nous obtenons la classification complète des surfaces hermitiennes localement homogènes qui satisfont la condition *-Einstein précédente. Nous construisons aussi des exemples de métriques hermitiennes non homogènes qui sont *-Einstein (mais non Einstein) sur 2 ¯ 2 , 1 × 1 et sur le produit d’une courbe de genre supérieur à 0 et d’une courbe de genre supérieur à 1.

@article{AIF_1999__49_5_1673_0,
     author = {Apostolov, Vestislav and Mu\v{s}karov, Oleg},
     title = {Weakly-Einstein hermitian surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {1673--1692},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {49},
     number = {5},
     year = {1999},
     doi = {10.5802/aif.1734},
     zbl = {0937.53035},
     mrnumber = {2000h:53091},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1734/}
}
TY  - JOUR
TI  - Weakly-Einstein hermitian surfaces
JO  - Annales de l'Institut Fourier
PY  - 1999
DA  - 1999///
SP  - 1673
EP  - 1692
VL  - 49
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1734/
UR  - https://zbmath.org/?q=an%3A0937.53035
UR  - https://www.ams.org/mathscinet-getitem?mr=2000h:53091
UR  - https://doi.org/10.5802/aif.1734
DO  - 10.5802/aif.1734
LA  - en
ID  - AIF_1999__49_5_1673_0
ER  - 
%0 Journal Article
%T Weakly-Einstein hermitian surfaces
%J Annales de l'Institut Fourier
%D 1999
%P 1673-1692
%V 49
%N 5
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.1734
%R 10.5802/aif.1734
%G en
%F AIF_1999__49_5_1673_0
Apostolov, Vestislav; Muškarov, Oleg. Weakly-Einstein hermitian surfaces. Annales de l'Institut Fourier, Volume 49 (1999) no. 5, pp. 1673-1692. doi : 10.5802/aif.1734. https://aif.centre-mersenne.org/articles/10.5802/aif.1734/

[1] V. Apostolov, J. Davidov and O. Muškarov, Self-dual hermitian surfaces, Trans. Amer. Math. Soc., 349 (1986), 3051-3063. | Zbl: 0880.53053

[2] V. Apostolov and P. Gauduchon, The Riemannian Goldberg-Sachs theorem, Int. J. Math., 8 (1997) 421-439. | MR: 98g:53080 | Zbl: 0891.53054

[3] T. Aubin, Equations du type Monge-Ampère sur les variétés kählériennes compactes. C.R.A.S. Paris, 283A (1976) 119. | MR: 55 #6496 | Zbl: 0333.53040

[4] T. Aubin, Nonlinear analysis on manifolds. Monge-Ampère Equations, Grund. Math. Wiss. 252, Springer, Berlin-Heildelberg-New York, 1982. | MR: 85j:58002 | Zbl: 0512.53044

[5] W. Barth, C. Peters and A. Van De Ven, Compact complex surfaces, Springer-Verlagh, Berlin-Heidelberg-New York-Tokyo, 1984. | MR: 86c:32026 | Zbl: 0718.14023

[6] F.A. Belgun, On the metric structure of non-Kähler complex surfaces, Preprint of Ecole Polytechnique (1998). | Zbl: 0988.32017

[7] A. Besse, Géométrie rieamannienne en dimension 4, Séminaire A.Besse, 1978-1979, eds. Bérard-Bergery, Berger, Houzel, CEDIC/Fernand Nathan, 1981.

[8] A. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb., Vol. 10, Springer, Berlin-Heildelberg-New-York, 1987. | MR: 88f:53087 | Zbl: 0613.53001

[9] C. Boyer, Conformal duality and compact complex surfaces, Math. Ann., 274 (1986), 517-526. | MR: 87i:53068 | Zbl: 0571.32017

[10] A. Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math., 49 (1983), 405-433. | Numdam | MR: 84h:53060 | Zbl: 0527.53030

[11] P. Gauduchon, Fibrés hermitiens à endomorphisme de Ricci non négatif, Bull. Soc. Math. France, 105 (1977), 113-140. | Numdam | MR: 58 #6375 | Zbl: 0382.53045

[12] P. Gauduchon, La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann., 267 (1984), 495-518. | MR: 87a:53101 | Zbl: 0536.53066

[13] P. Gauduchon, Structures de Weyl-Einstein, espaces de twisteurs et variétés de type S1 ȕ S3, J. reine angew. Math., 469 (1995), 1-50. | MR: 97d:53048 | Zbl: 0858.53039

[14] P. Gauduchon and L. Ornea, Locally conformally Kähler metrics on Hopf surfaces, Ann. Inst. Fourier, 48-4 (1998), 1107-1127. | Numdam | MR: 2000g:53088 | Zbl: 0917.53025

[15] G. Grantcharov and O. Muškarov, Hermitian *-Einstein surfaces, Proc. Amer. Math. Soc. 120 (1994), 233-239. | MR: 94b:53081 | Zbl: 0796.53068

[16] G.R. Jensen, Homogeneous Einstein spaces of dimension 4, J. Differential Geom., 3 (1969), 309-349. | MR: 41 #6100 | Zbl: 0194.53203

[17] J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math., 99 (1974), 203-219. | MR: 49 #7949 | Zbl: 0278.53031

[18] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry I, II, Interscience Publishers, 1963. | Zbl: 0119.37502

[19] C. Lebrun, Einstein Metrics on Complex Surfaces, in Geometry and Physics (Aarhus 1995), Eds. J. Andersen, J. Dupont, H. Pedersen and A. Swann, Lect. Notes in Pure Appl. Math., Marcel Dekker, 1996.

[20] P. Nurowski, Einstein equations and Cauchy-Riemann geometry, Ph. D. Thesis, SISSA/ISAS, Trieste (1993).

[21] D. Page, A compact rotating Gravitational Instanton, Phys. Lett., 79 B (1979), 235-238.

[22] L. Ornea and P. Piccinni, Induced Hopf bundles and Einstein metrics, in New developments in differential geometry, Budapest 1996, 295-305, Kluwer Acad. Publ., Dordrecht, 1999. | Zbl: 0949.53033

[23] M. Pontecorvo, Uniformization of conformally flat Hermitian surfaces, Diff. Geom. and its Appl., 2 (1992), 295-305. | MR: 94k:32052 | Zbl: 0766.53052

[24] M. Przanowski and B. Broda, Locally Kähler Gravitational Instantons, Acta Phys. Pol., B14 (1983), 637-661.

[25] Y.T. Siu, Every K3 surface is Kähler, Invent. Math., 73 (1983), 139-150. | MR: 84j:32036 | Zbl: 0557.32004

[26] G. Tian, On Calabi's Conjecture for Complex Surfaces with positive First Chern Class, Invent. Math., 10, (1990), 101-172. | MR: 91d:32042 | Zbl: 0716.32019

[27] K.P. Tod, Cohomogeneity-One Metrics with Self-dual Weyl Tensor, Twistor Theory (S. Huggett, ed.), Marcel Dekker Inc., New York, 1995, pp. 171-184. | MR: 95i:53056 | Zbl: 0827.53017

[28] A. Todorov, Applications of the Kähler-Einstein Calabi-Yau metrics to moduli of K3 surfaces, Invent. Math., 61 (1980), 251-265. | MR: 82k:32065 | Zbl: 0472.14006

[29] F. Tricerri and L. Vanhecke, Curvature tensors on almost-Hermitian manifolds, Trans. Amer. Math. Soc., 267 (1981), 365-398. | MR: 82j:53071 | Zbl: 0484.53014

[30] I. Vaisman, On locally and globally conformal Kähler manifolds, Trans. Amer. Math. Soc., 262 (1980), 533-542. | MR: 81j:53064 | Zbl: 0446.53048

[31] I. Vaisman, Generalized Hopf manifolds, Geom. Dedicata, 13 (1982), 231-255. | MR: 84g:53096 | Zbl: 0506.53032

[32] I. Vaisman, Some curvature properties of complex surfaces, Ann. Mat. Pura Appl., 32 (1982), 1-18. | MR: 84i:53064 | Zbl: 0512.53058

[33] S-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Comm. Pure Appl. Math., 31 (1978) 339-411. | MR: 81d:53045 | Zbl: 0369.53059

Cited by Sources: