Hermitian spin surfaces with small eigenvalues of the Dolbeault operator

Bogdan Alexandrov[1]

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

Annales de l'Institut Fourier (2004)

  • Volume: 54, Issue: 7, page 2437-2453
  • ISSN: 0373-0956

Abstract

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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

How to cite

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Alexandrov, Bogdan. "Hermitian spin surfaces with small eigenvalues of the Dolbeault operator." Annales de l'Institut Fourier 54.7 (2004): 2437-2453. <http://eudml.org/doc/116178>.

@article{Alexandrov2004,
abstract = {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},
affiliation = {Universität Greifswald, Institut für Mathemathik und Informatik, Friedrich-Ludwig-Jahn-Str. 15a, 17487 Greifswald (Allemagne)},
author = {Alexandrov, Bogdan},
journal = {Annales de l'Institut Fourier},
keywords = {Hermitian surface; locally conformally Kähler metric; ruled surface; Hopf surface; Kähler metric; locally conformal Kähler metric},
language = {eng},
number = {7},
pages = {2437-2453},
publisher = {Association des Annales de l'Institut Fourier},
title = {Hermitian spin surfaces with small eigenvalues of the Dolbeault operator},
url = {http://eudml.org/doc/116178},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Alexandrov, Bogdan
TI - Hermitian spin surfaces with small eigenvalues of the Dolbeault operator
JO - Annales de l'Institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 7
SP - 2437
EP - 2453
AB - 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
LA - eng
KW - Hermitian surface; locally conformally Kähler metric; ruled surface; Hopf surface; Kähler metric; locally conformal Kähler metric
UR - http://eudml.org/doc/116178
ER -

References

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  1. B. Alexandrov, G. Grantcharov, S. Ivanov, The Dolbeault operator on Hermitian spin surfaces, Ann. Inst. Fourier 51 (2001), 221-235 Zbl0987.53011MR1821075
  2. C. Bär, Real Killing spinors and holonomy, Comm. Math. Phys 154 (1993), 509-521 Zbl0778.53037MR1224089
  3. W. Barth, C. Peters, A. Van de Ven, Compact Complex Surfaces, (1984), Springer-Verlag Zbl0718.14023MR749574
  4. A. Beauville, Complex algebraic surfaces, (1983), Cambridge University Press Zbl0512.14020MR732439
  5. F. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann 317 (2000), 1-40 Zbl0988.32017MR1760667
  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) 40 (1976), 273-288 Zbl0352.32020MR427325
  7. T. Friedrich, Der erste Eigenwert des Dirac--Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr 97 (1980), 117-146 Zbl0462.53027MR600828
  8. T. Friedrich, The classification of,dimensional Kähler manifolds with small eigenvalue of the Dirac operator, Math. Ann 295 (1993), 565-574 Zbl0798.53065MR1204838
  9. P. Gauduchon, Le théorème de l'excentricité nulle, C. R. Acad. Sci. Paris Ser. A 285 (1977), 387-390 Zbl0362.53024MR470920
  10. P. Gauduchon, Fibrés hermitiens à endomorphisme de Ricci non négatif, Bul. Soc. Math. France 105 (1977), 113-140 Zbl0382.53045MR486672
  11. P. Gauduchon, Surfaces de Hopf - variétés presque-complexes de dimension quatre, Géométrie riemannienne en dimension 4. Semin. Arthur Besse, Paris 1978/79 (1981), 134-155 Zbl0513.53057
  12. P. Gauduchon, La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann 267 (1984), 495-518 Zbl0523.53059MR742896
  13. P. Gauduchon, Hermitian connections and Dirac operators, Bol. U. M. I. ser. VII XI-B, supl. 2 (1997), 257-289 Zbl0876.53015MR1456265
  14. P. Gauduchon, L. Ornea, Locally conformally Kähler metrics on Hopf surfaces, Ann. Inst. Fourier 48 (1998), 1107-1127 Zbl0917.53025MR1656010
  15. M. Gromov, H. B. Lawson, Spin and scalar curvature in the presence of a fundamental group I, Ann. Math 111 (1980), 209-230 Zbl0445.53025MR569070
  16. R. Hartshorne, Algebraic geometry, 52 (1977), Springer-Verlag Zbl0367.14001MR463157
  17. N. Hitchin, Harmonic spinors, Adv. Math 14 (1974), 1-55 Zbl0284.58016MR358873
  18. M. Inoue, On Surfaces of Class V I I 0 , Invent. Math. 24 (1974), 269-310 Zbl0283.32019MR342734
  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 4 (1986), 291-325 Zbl0629.53058MR910548
  20. K.-D. Kirchberg, The first eigenvalue of the Dirac operator on Kähler manifolds, J. Geom. Phys 7 (1990), 447-468 Zbl0734.53050MR1131907
  21. K. Kodaira, On the structure of compact analytic spaces I, Am. J. Math 86 (1964), 751-798 Zbl0137.17501MR187255
  22. K. Kodaira, On the structure of compact analytic spaces II, Am. J. Math 88 (1966), 682-721 Zbl0193.37701MR205280
  23. K. Kodaira, On the structure of compact analytic spaces III, Am. J. Math 90 (1969), 55-83 MR228019
  24. K. Kodaira, D. C. Spencer, On the variation of almost-complex structure, Princeton Math. Ser 12 (1957), 139-150 Zbl0082.15402MR88775
  25. H. B. Lawson, M.-L. Michelsohn, Spin geometry, 38 (1989), Princeton Univ. Press, Princeton Zbl0688.57001MR1031992
  26. K. Tsukada, Holomorphic forms and holomorphic vector fields on compact generalized Hopf manifolds, Compos. Math 93 (1994), 1-22 Zbl0811.53032MR1286795
  27. I. Vaisman, On locally and globally conformally Kähler manifolds, Trans. Am. Math. Soc 262 (1980), 533-542 Zbl0446.53048MR586733
  28. I. Vaisman, Some curvature properties of complex surfaces, Ann. Mat. Pura Appl 132 (1982), 231-255 Zbl0512.53058MR696036
  29. S.-T. Yau, On the curvature of compact Hermitian manifolds, Invent. Math 25 (1974), 213-239 Zbl0299.53039MR382706

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