Hermitian spin surfaces with small eigenvalues of the Dolbeault operator
[Surfaces hérmitiennes de spin avec des petites valeurs propres pour l'opérateur de Dolbeault]
Alexandrov, Bogdan
Annales de l'Institut Fourier, Tome 54 (2004), p. 2437-2453 / Harvested from Numdam
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Nous étudions les variétés hermitiennes de spin avec courbure scalaire conforme positive sur lesquelles la première valeur propre de l'opérateur de Dolbeault est la plus petite possible. On montre qu'une telle surface est une surface réglée, ou bien une surface de Hopf. Nous donnons une classification complète des surfaces réglées avec cette propriété. Pour les surfaces de Hopf on obtient une classification partielle et quelques exemples.

We study the compact Hermitian spin surfaces with positive conformal scalar curvature on which the first eigenvalue of the Dolbeault operator of the spin structure is the smallest possible. We prove that such a surface is either a ruled surface or a Hopf surface. We give a complete classification of the ruled surfaces with this property. For the Hopf surfaces we obtain a partial classification and some examples

Publié le : 2004-01-01
DOI : https://doi.org/10.5802/aif.2085
Classification:  53C55,  32J15
Mots clés: surface hermitienne, métrique localement conformément Kählérienne, surface réglée, surface de Hopf 
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     author = {Alexandrov, Bogdan},
     title = {Hermitian spin surfaces with small eigenvalues of the Dolbeault operator},
     journal = {Annales de l'Institut Fourier},
     volume = {54},
     year = {2004},
     pages = {2437-2453},
     doi = {10.5802/aif.2085},
     mrnumber = {2139699},
     zbl = {1083.53067},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2004__54_7_2437_0}
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Alexandrov, Bogdan. Hermitian spin surfaces with small eigenvalues of the Dolbeault operator. Annales de l'Institut Fourier, Tome 54 (2004) pp. 2437-2453. doi : 10.5802/aif.2085. http://gdmltest.u-ga.fr/item/AIF_2004__54_7_2437_0/

[1] B. Alexandrov; G. Grantcharov; S. Ivanov The Dolbeault operator on Hermitian spin surfaces, Ann. Inst. Fourier, Tome 51 (2001) no. 1, pp. 221-235 | Article | Numdam | MR 1821075 | Zbl 0987.53011

[2] C. Bär Real Killing spinors and holonomy, Comm. Math. Phys, Tome 154 (1993), pp. 509-521 | Article | MR 1224089 | Zbl 0778.53037

[3] W. Barth; C. Peters; A. Van De Ven Compact Complex Surfaces, Springer-Verlag (1984) | MR 749574 | Zbl 0718.14023

[4] A. Beauville Complex algebraic surfaces, Cambridge University Press (1983) | MR 732439 | Zbl 0512.14020

[5] F. Belgun On the metric structure of non-Kähler complex surfaces, Math. Ann, Tome 317 (2000), pp. 1-40 | Article | MR 1760667 | Zbl 0988.32017

[6] F.A. Bogomolov Classification of surfaces of class VII 0 with b 2 =0, Izv. Akad. Nauk SSSR, Ser. Mat,(in Russian), Tome 40 (1976), pp. 273-288 | MR 427325 | Zbl 0352.32020

[7] T. Friedrich Der erste Eigenwert des Dirac--Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr, Tome 97 (1980), pp. 117-146 | Article | MR 600828 | Zbl 0462.53027

[8] T. Friedrich The classification of,dimensional Kähler manifolds with small eigenvalue of the Dirac operator, Math. Ann, Tome 295 (1993), pp. 565-574 | Article | MR 1204838 | Zbl 0798.53065

[9] P. Gauduchon Le théorème de l'excentricité nulle, C. R. Acad. Sci. Paris Ser. A, Tome 285 (1977), pp. 387-390 | MR 470920 | Zbl 0362.53024

[10] P. Gauduchon Fibrés hermitiens à endomorphisme de Ricci non négatif, Bul. Soc. Math. France, Tome 105 (1977), pp. 113-140 | Numdam | MR 486672 | Zbl 0382.53045

[11] P. Gauduchon Surfaces de Hopf - variétés presque-complexes de dimension quatre, Géométrie riemannienne en dimension 4. Semin. Arthur Besse, Paris 1978/79 (1981), pp. 134-155 | Zbl 0513.53057

[12] P. Gauduchon La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann, Tome 267 (1984), pp. 495-518 | Article | MR 742896 | Zbl 0523.53059

[13] P. Gauduchon Hermitian connections and Dirac operators, Bol. U. M. I. ser. VII, Tome XI-B, supl. 2 (1997), pp. 257-289 | MR 1456265 | Zbl 0876.53015

[14] P. Gauduchon; L. Ornea Locally conformally Kähler metrics on Hopf surfaces, Ann. Inst. Fourier, Tome 48 (1998), pp. 1107-1127 | Article | Numdam | MR 1656010 | Zbl 0917.53025

[15] M. Gromov; H. B. Lawson Spin and scalar curvature in the presence of a fundamental group I, Ann. Math, Tome 111 (1980), pp. 209-230 | Article | MR 569070 | Zbl 0445.53025

[16] R. Hartshorne Algebraic geometry, Springer-Verlag, Graduate Texts in Mathematics, Tome 52 (1977) | MR 463157 | Zbl 0367.14001

[17] N. Hitchin Harmonic spinors, Adv. Math, Tome 14 (1974), pp. 1-55 | Article | MR 358873 | Zbl 0284.58016

[18] M. Inoue On Surfaces of Class VII 0 , Invent. Math., Tome 24 (1974), pp. 269-310 | Article | MR 342734 | Zbl 0283.32019

[19] K.-D. Kirchberg An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature, Ann. Glob. Anal. Geom, Tome 4 (1986), pp. 291-325 | Article | MR 910548 | Zbl 0629.53058

[20] K.-D. Kirchberg The first eigenvalue of the Dirac operator on Kähler manifolds, J. Geom. Phys, Tome 7 (1990), pp. 447-468 | MR 1131907 | Zbl 0734.53050

[21] K. Kodaira On the structure of compact analytic spaces I, Am. J. Math, Tome 86 (1964), pp. 751-798 | Article | MR 187255 | Zbl 0137.17501

[22] K. Kodaira On the structure of compact analytic spaces II, Am. J. Math, Tome 88 (1966), pp. 682-721 | Article | MR 205280 | Zbl 0193.37701

[23] K. Kodaira On the structure of compact analytic spaces III, Am. J. Math, Tome 90 (1969), pp. 55-83 | Article | MR 228019

[24] K. Kodaira; D. C. Spencer On the variation of almost-complex structure, Princeton Math. Ser, Tome 12 (1957), pp. 139-150 | MR 88775 | Zbl 0082.15402

[25] H. B. Lawson; M.-L. Michelsohn Spin geometry, Princeton Univ. Press, Princeton, Princeton Mathematical Series, Tome 38 (1989) | MR 1031992 | Zbl 0688.57001

[26] K. Tsukada Holomorphic forms and holomorphic vector fields on compact generalized Hopf manifolds, Compos. Math, Tome 93 (1994) no. 1, pp. 1-22 | Numdam | MR 1286795 | Zbl 0811.53032

[27] I. Vaisman On locally and globally conformally Kähler manifolds, Trans. Am. Math. Soc, Tome 262 (1980), pp. 533-542 | MR 586733 | Zbl 0446.53048

[27] S.-T. Yau On the curvature of compact Hermitian manifolds, Invent. Math, Tome 25 (1974), pp. 213-239 | Article | MR 382706 | Zbl 0299.53039

[28] I. Vaisman Some curvature properties of complex surfaces, Ann. Mat. Pura Appl, Tome 132 (1982), pp. 231-255 | MR 696036 | Zbl 0512.53058