Hermitian spin surfaces with small eigenvalues of the Dolbeault operator

Alexandrov, Bogdan

doi:10.5802/aif.2085

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 ¹

¹ Universität Greifswald, Institut für Mathemathik und Informatik, Friedrich-Ludwig-Jahn-Str. 15a, 17487 Greifswald (Allemagne)

Annales de l'Institut Fourier, Tome 54 (2004) no. 7, pp. 2437-2453.

Résumé
Abstract

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

MR Zbl

DOI : 10.5802/aif.2085

Classification : 53C55, 32J15
Keywords: Hermitian surface, locally conformally Kähler metric, ruled surface, Hopf surface
Mot clés : surface hermitienne, métrique localement conformément Kählérienne, surface réglée, surface de Hopf

Affiliations des auteurs :

Alexandrov, Bogdan ¹

¹ Universität Greifswald, Institut für Mathemathik und Informatik, Friedrich-Ludwig-Jahn-Str. 15a, 17487 Greifswald (Allemagne)

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     title = {Hermitian spin surfaces with small eigenvalues of the {Dolbeault} operator},
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     volume = {54},
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Alexandrov, Bogdan. Hermitian spin surfaces with small eigenvalues of the Dolbeault operator. Annales de l'Institut Fourier, Tome 54 (2004) no. 7, pp. 2437-2453. doi : 10.5802/aif.2085. http://www.numdam.org/articles/10.5802/aif.2085/

Bibliographie
Cité par

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

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

[3] W. Barth; C. Peters; A. Van de Ven Compact Complex Surfaces, Springer-Verlag, 1984 | MR | Zbl

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

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

[6] F.A. Bogomolov Classification of surfaces of class $V I I_{0}$ with $b_{2} = 0$ , Izv. Akad. Nauk SSSR, Ser. Mat,(in Russian), Volume 40 (1976), pp. 273-288 | MR | Zbl

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

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

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

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

[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

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

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

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

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

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

[17] N. Hitchin Harmonic spinors, Adv. Math, Volume 14 (1974), pp. 1-55 | DOI | MR | Zbl

[18] M. Inoue On Surfaces of Class $V I I_{0}$ , Invent. Math., Volume 24 (1974), pp. 269-310 | DOI | MR | Zbl

[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, Volume 4 (1986), pp. 291-325 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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