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
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.
Keywords: Hermitian surface, locally conformally Kähler metric, ruled surface, Hopf surface
Mot clés : surface hermitienne, métrique localement conformément Kählérienne, surface réglée, surface de Hopf
Alexandrov, Bogdan 1
@article{AIF_2004__54_7_2437_0, author = {Alexandrov, Bogdan}, title = {Hermitian spin surfaces with small eigenvalues of the {Dolbeault} operator}, journal = {Annales de l'Institut Fourier}, pages = {2437--2453}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {54}, number = {7}, year = {2004}, doi = {10.5802/aif.2085}, zbl = {1083.53067}, mrnumber = {2139699}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2085/} }
TY - JOUR AU - Alexandrov, Bogdan TI - Hermitian spin surfaces with small eigenvalues of the Dolbeault operator JO - Annales de l'Institut Fourier PY - 2004 SP - 2437 EP - 2453 VL - 54 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2085/ DO - 10.5802/aif.2085 LA - en ID - AIF_2004__54_7_2437_0 ER -
%0 Journal Article %A Alexandrov, Bogdan %T Hermitian spin surfaces with small eigenvalues of the Dolbeault operator %J Annales de l'Institut Fourier %D 2004 %P 2437-2453 %V 54 %N 7 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2085/ %R 10.5802/aif.2085 %G en %F AIF_2004__54_7_2437_0
Alexandrov, Bogdan. Hermitian spin surfaces with small eigenvalues of the Dolbeault operator. Annales de l'Institut Fourier, Volume 54 (2004) no. 7, pp. 2437-2453. doi : 10.5802/aif.2085. https://aif.centre-mersenne.org/articles/10.5802/aif.2085/
[1] The Dolbeault operator on Hermitian spin surfaces, Ann. Inst. Fourier, Volume 51 (2001) no. 1, pp. 221-235 | DOI | Numdam | MR | Zbl
[2] Real Killing spinors and holonomy, Comm. Math. Phys, Volume 154 (1993), pp. 509-521 | DOI | MR | Zbl
[3] Compact Complex Surfaces, Springer-Verlag, 1984 | MR | Zbl
[4] Complex algebraic surfaces, Cambridge University Press, 1983 | MR | Zbl
[5] On the metric structure of non-Kähler complex surfaces, Math. Ann, Volume 317 (2000), pp. 1-40 | DOI | MR | Zbl
[6] Classification of surfaces of class with , Izv. Akad. Nauk SSSR, Ser. Mat,(in Russian), Volume 40 (1976), pp. 273-288 | MR | Zbl
[7] Der erste Eigenwert des Dirac--Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr, Volume 97 (1980), pp. 117-146 | DOI | MR | Zbl
[8] 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] Le théorème de l'excentricité nulle, C. R. Acad. Sci. Paris Ser. A, Volume 285 (1977), pp. 387-390 | MR | Zbl
[10] Fibrés hermitiens à endomorphisme de Ricci non négatif, Bul. Soc. Math. France, Volume 105 (1977), pp. 113-140 | Numdam | MR | Zbl
[11] 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] La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann, Volume 267 (1984), pp. 495-518 | DOI | MR | Zbl
[13] Hermitian connections and Dirac operators, Bol. U. M. I. ser. VII, Volume XI-B, supl. 2 (1997), pp. 257-289 | MR | Zbl
[14] Locally conformally Kähler metrics on Hopf surfaces, Ann. Inst. Fourier, Volume 48 (1998), pp. 1107-1127 | DOI | Numdam | MR | Zbl
[15] Spin and scalar curvature in the presence of a fundamental group I, Ann. Math, Volume 111 (1980), pp. 209-230 | DOI | MR | Zbl
[16] Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, 1977 | MR | Zbl
[17] Harmonic spinors, Adv. Math, Volume 14 (1974), pp. 1-55 | DOI | MR | Zbl
[18] On Surfaces of Class , Invent. Math., Volume 24 (1974), pp. 269-310 | DOI | MR | Zbl
[19] 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] The first eigenvalue of the Dirac operator on Kähler manifolds, J. Geom. Phys, Volume 7 (1990), pp. 447-468 | MR | Zbl
[21] On the structure of compact analytic spaces I, Am. J. Math, Volume 86 (1964), pp. 751-798 | DOI | MR | Zbl
[22] On the structure of compact analytic spaces II, Am. J. Math, Volume 88 (1966), pp. 682-721 | DOI | MR | Zbl
[23] On the structure of compact analytic spaces III, Am. J. Math, Volume 90 (1969), pp. 55-83 | DOI | MR
[24] On the variation of almost-complex structure, Princeton Math. Ser, Volume 12 (1957), pp. 139-150 | MR | Zbl
[25] Spin geometry, Princeton Mathematical Series, 38, Princeton Univ. Press, Princeton, 1989 | MR | Zbl
[26] 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] On locally and globally conformally Kähler manifolds, Trans. Am. Math. Soc, Volume 262 (1980), pp. 533-542 | MR | Zbl
[27] On the curvature of compact Hermitian manifolds, Invent. Math, Volume 25 (1974), pp. 213-239 | DOI | MR | Zbl
[28] Some curvature properties of complex surfaces, Ann. Mat. Pura Appl, Volume 132 (1982), pp. 231-255 | MR | Zbl
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