The Dolbeault operator on Hermitian spin surfaces
[L'opérateur de Dolbeault sur les surfaces hermitiennes de spin]
Alexandrov, Bodgan1 ; Grantcharov, Gueo2 ; Ivanov, Stefan2
1 Humboldt University, Institute for Mathematics, Rudower Chaussee 25, 10099 Berlin (Allemagne)
2 University of Sofia, Faculty of Mathematics and Informatics, Department of Geometry, 5 James Bourchier Blvd, 1126 Sofia (Bulgarie)
Annales de l'Institut Fourier, Tome 51 (2001) no. 1, pp. 221-235.

On démontre l'annulation du noyau de l'opérateur de Dolbeault sur la racine carrée du fibré canonique d'une surface hermitienne de spin avec courbure scalaire positive. On obtient des minorations pour la première valeur propre de cet opérateur, dans le cas où la courbure scalaire conforme est non-négative.

We prove the vanishing of the kernel of the Dolbeault operator of the square root of the canonical line bundle of a compact Hermitian spin surface with positive scalar curvature. We give lower estimates of the eigenvalues of this operator when the conformal scalar curvature is non -negative.

DOI : 10.5802/aif.1822
Classification : 53C15, 53C25, 53B35
Keywords: hermitian surfaces, Dirac operator, Dolbeault operator, twistor spinors
Mot clés : surfaces hermitiennes, opérateur de Dirac, opérateur de Dolbeault, spineurs twisteurs
Alexandrov, Bodgan 1 ; Grantcharov, Gueo 2 ; Ivanov, Stefan 2

1 Humboldt University, Institute for Mathematics, Rudower Chaussee 25, 10099 Berlin (Allemagne)
2 University of Sofia, Faculty of Mathematics and Informatics, Department of Geometry, 5 James Bourchier Blvd, 1126 Sofia (Bulgarie) 
@article{AIF_2001__51_1_221_0,
     author = {Alexandrov, Bodgan and Grantcharov, Gueo and Ivanov, Stefan},
     title = {The {Dolbeault} operator on {Hermitian} spin surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {221--235},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {51},
     number = {1},
     year = {2001},
     doi = {10.5802/aif.1822},
     mrnumber = {1821075},
     zbl = {0987.53011},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1822/}
}
TY  - JOUR
AU  - Alexandrov, Bodgan
AU  - Grantcharov, Gueo
AU  - Ivanov, Stefan
TI  - The Dolbeault operator on Hermitian spin surfaces
JO  - Annales de l'Institut Fourier
PY  - 2001
SP  - 221
EP  - 235
VL  - 51
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1822/
DO  - 10.5802/aif.1822
LA  - en
ID  - AIF_2001__51_1_221_0
ER  - 
%0 Journal Article
%A Alexandrov, Bodgan
%A Grantcharov, Gueo
%A Ivanov, Stefan
%T The Dolbeault operator on Hermitian spin surfaces
%J Annales de l'Institut Fourier
%D 2001
%P 221-235
%V 51
%N 1
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1822/
%R 10.5802/aif.1822
%G en
%F AIF_2001__51_1_221_0
Alexandrov, Bodgan; Grantcharov, Gueo; Ivanov, Stefan. The Dolbeault operator on Hermitian spin surfaces. Annales de l'Institut Fourier, Tome 51 (2001) no. 1, pp. 221-235. doi : 10.5802/aif.1822. http://www.numdam.org/articles/10.5802/aif.1822/

[1] V. Apostolov; J. Davidov; O. Mu¡karov Self-dual Hermitian surfaces, Trans. Amer. Math. Soc., Volume 349 (1996), pp. 3051-3063 | DOI | Zbl

[2] N. Aronszjan A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl., Volume 35 (1957), pp. 235-249 | MR | Zbl

[3] M.F. Atiyah; R. Bott; A. Shapiro Clifford modules, Topology, Volume 3 (Suppl.1) (1964), pp. 3-38 | DOI | MR | Zbl

[4] H. Baum; T. Friedrich; R. Grunewald; I. Kath Twistor and Killing spinors on Riemannian manifolds (Seminarbericht), Volume 108 (1990) | Zbl

[5] J.-M. Bismut A local index theorem for non-Kähler manifolds, Math. Ann., Volume 284 (1989), pp. 681-699 | DOI | MR | Zbl

[6] C. Boyer A note on hyperhermitian four manifolds, Proc. Amer. Math. Soc., Volume 102 (1988) no. 1, pp. 157-164 | MR | Zbl

[7] T. Friedrich Der erste Eigenwert des Dirac operators einer kompakten Riemannischen Manningfaltigkeit nichtnegativer Skalarkrümung, Math. Nachrichten, Volume 97 (1980), pp. 117-146 | DOI | MR | Zbl

[8] T. Friedrich The classification of 4-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, Sér. 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 La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann., Volume 267 (1984), pp. 495-518 | DOI | MR | Zbl

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

[13] O. Hijazi Opérateurs de Dirac sur le variétés riemanniennes : minoration des valeurs propres (1984) (Thèse de 3e Cycle, École Polytechnique)

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

[15] 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

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

[17] K.-D. Kirchberg Properties of Kählerian twistor-spinors and vanishing theorems, Math. Ann., Volume 293 (1992), pp. 349-369 | DOI | MR | Zbl

[18] A. Moroianu Structures de Weyl admettant des spineurs parallèles, Bull. Soc. Math. France, Volume 124 (1996) no. 4, pp. 685-695 | Numdam | MR | Zbl

[19] M. Pontecorvo Complex structures on Riemannian four-manifolds, Math. Ann., Volume 309 (1997), pp. 159-177 | DOI | MR | Zbl

[20] I. Vaisman Some curvature properties of complex surfaces, Ann. Mat. Pura Appl., Volume 132 (1982), pp. 1-18 | DOI | MR | Zbl

[21] M. Wang Parallel spinors and parallel forms, Ann. Glob. Anal. Geom., Volume 7 (1989), pp. 59-68 | DOI | MR | Zbl

Cité par Sources :