[Fibrés homogènes et première valeur propre sur les espaces symétriques]
On montre que le point de Gieseker d’un fibré homogène irréductible sur un espace homogène rationnel est stable. On en déduit une majoration optimale de la première valeur propre du laplacien d’une métrique Kählérienne quelconque sur un espace symétrique Hermitien compact du type ABDC.
In this note we prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kähler metric on a compact Hermitian symmetric spaces of ABCD–type.
Keywords: Homogeneous bundles, spectrum of the Laplacian
Mot clés : fibrés homogènes, spectre du Laplacien
@article{AIF_2008__58_7_2315_0,
author = {Biliotti, Leonardo and Ghigi, Alessandro},
title = {Homogeneous bundles and the first eigenvalue of symmetric spaces},
journal = {Annales de l'Institut Fourier},
pages = {2315--2331},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {58},
number = {7},
year = {2008},
doi = {10.5802/aif.2415},
mrnumber = {2498352},
zbl = {1161.53064},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2415/}
}
TY - JOUR AU - Biliotti, Leonardo AU - Ghigi, Alessandro TI - Homogeneous bundles and the first eigenvalue of symmetric spaces JO - Annales de l'Institut Fourier PY - 2008 SP - 2315 EP - 2331 VL - 58 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2415/ DO - 10.5802/aif.2415 LA - en ID - AIF_2008__58_7_2315_0 ER -
%0 Journal Article %A Biliotti, Leonardo %A Ghigi, Alessandro %T Homogeneous bundles and the first eigenvalue of symmetric spaces %J Annales de l'Institut Fourier %D 2008 %P 2315-2331 %V 58 %N 7 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2415/ %R 10.5802/aif.2415 %G en %F AIF_2008__58_7_2315_0
Biliotti, Leonardo; Ghigi, Alessandro. Homogeneous bundles and the first eigenvalue of symmetric spaces. Annales de l'Institut Fourier, Tome 58 (2008) no. 7, pp. 2315-2331. doi : 10.5802/aif.2415. https://aif.centre-mersenne.org/articles/10.5802/aif.2415/
[1] Lie group actions in complex analysis, Aspects of Mathematics, E27, Friedr. Vieweg & Sohn, Braunschweig, 1995 | MR | Zbl
[2] Stable bundles and the first eigenvalue of the Laplacian, J. Geom. Anal., Volume 17 (2007) no. 3, pp. 375-386 | MR | Zbl
[3] The Penrose transform, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1989 | MR | Zbl
[4] Upper bound for the first eigenvalue of algebraic submanifolds, Comment. Math. Helv., Volume 69 (1994) no. 2, pp. 199-207 | DOI | MR | Zbl
[5] Riemannian metrics with large , Proc. Amer. Math. Soc., Volume 122 (1994) no. 3, pp. 905-906 | MR | Zbl
[6] The geometry of four-manifolds, Oxford Mathematical Monographs. Oxford: Clarendon Press. ix, 440p., New York, 1990 | MR | Zbl
[7] Riemannian manifolds admitting isometric immersions by their first eigenfunctions, Pacific J. Math., Volume 195 (2000) no. 1, pp. 91-99 | DOI | MR | Zbl
[8] Cycle spaces of flag domains, Progress in Mathematics, 245, Birkhäuser Boston Inc., Boston, MA, 2006 (A complex geometric viewpoint) | MR | Zbl
[9] Kähler-Einstein metrics and integral invariants, Springer-Verlag, Berlin, 1988 | MR | Zbl
[10] On the moduli of vector bundles on an algebraic surface, Ann. of Math. (2), Volume 106 (1977) no. 1, pp. 45-60 | DOI | MR | Zbl
[11] Analytic Hilbert quotients, Several complex variables (Berkeley, CA, 1995-1996), Volume 37, Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge, 1999, pp. 309-349 | MR | Zbl
[12] Cartan decomposition of the moment map, Math. Ann., Volume 337 (2007) no. 1, pp. 197-232 | DOI | MR | Zbl
[13] Differential geometry, Lie groups, and symmetric spaces, 80, Pure and Applied Mathematic, Academic Press Inc., XV. 628 p., New York, 1978 | MR | Zbl
[14] Introduction to Lie algebras and representation theory, 9, Graduate Texts in Mathematics, Springer-Verlag, New York, 1978 (Second printing, revised) | MR | Zbl
[15] The length of vectors in representation spaces, Algebraic geometry. (Proc. Summer Meeting, Copenhagen, 1978), Volume 732, Lecture Notes in Math., Springer, Berlin, 1979, pp. 233-243 | MR | Zbl
[16] Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, Volume 15, Princeton University Press, Princeton, NJ, 1987 (Kanô Memorial Lectures, 5) | MR | Zbl
[17] On filtered Lie algebras and geometric structures. II, J. Math. Mech., Volume 14 (1965), pp. 513-521 | MR | Zbl
[18] Sur les orbites fermées des groupes algébriques réductifs, Invent. Math., Volume 16 (1972), pp. 1-5 | DOI | MR | Zbl
[19] Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34, Springer-Verlag, Berlin, 1994 (third edition) | MR | Zbl
[20] Lie groups and algebraic groups, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1990 (Translated from the Russian and with a preface by D. A. Leites) | MR | Zbl
[21] Spinor bundles on quadrics, Trans. Amer. Math. Soc., Volume 307 (1988) no. 1, pp. 301-316 | DOI | MR | Zbl
[22] Rational homogeneous varieties, Notes from a course held in Cortona, Italy, 1995 (http://www.math.unifi.it/ottavian/public.html)
[23] Holomorphic vector bundles on homogeneous spaces, Topology, Volume 5 (1966), pp. 159-177 | DOI | MR | Zbl
[24] On a theorem of Ramanan, Nagoya Math. J., Volume 69 (1978), pp. 131-138 | MR | Zbl
[25] Balance point and stability of vector bundles over a projective manifold, Math. Res. Lett., Volume 9(2-3) (2002), pp. 393-411 | DOI | MR | Zbl
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