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.
@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},
volume = {58},
year = {2008},
pages = {2315-2331},
doi = {10.5802/aif.2415},
zbl = {1161.53064},
mrnumber = {2498352},
language = {en},
url = {http://dml.mathdoc.fr/item/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) pp. 2315-2331. doi : 10.5802/aif.2415. http://gdmltest.u-ga.fr/item/AIF_2008__58_7_2315_0/
[1] Lie group actions in complex analysis, Friedr. Vieweg & Sohn, Braunschweig, Aspects of Mathematics, Tome E27 (1995) | MR 1334091 | Zbl 0845.22001
[2] Stable bundles and the first eigenvalue of the Laplacian, J. Geom. Anal., Tome 17 (2007) no. 3, pp. 375-386 | MR 2358762 | Zbl 1128.58013
[3] The Penrose transform, The Clarendon Press Oxford University Press, New York, Oxford Mathematical Monographs (1989) | MR 1038279 | Zbl 0726.58004
[4] Upper bound for the first eigenvalue of algebraic submanifolds, Comment. Math. Helv., Tome 69 (1994) no. 2, pp. 199-207 | Article | MR 1282367 | Zbl 0814.53040
[5] Riemannian metrics with large , Proc. Amer. Math. Soc., Tome 122 (1994) no. 3, p. 905-906 | MR 1213857 | Zbl 0820.58056
[6] The geometry of four-manifolds, Oxford Mathematical Monographs. Oxford: Clarendon Press. ix, 440p., New York (1990) | MR 1079726 | Zbl 0820.57002
[7] Riemannian manifolds admitting isometric immersions by their first eigenfunctions, Pacific J. Math., Tome 195 (2000) no. 1, pp. 91-99 | Article | MR 1781616 | Zbl 1030.53043
[8] Cycle spaces of flag domains, Birkhäuser Boston Inc., Boston, MA, Progress in Mathematics, Tome 245 (2006) (A complex geometric viewpoint) | MR 2188135 | Zbl 1084.22011
[9] Kähler-Einstein metrics and integral invariants, Springer-Verlag, Berlin (1988) | MR 947341 | Zbl 0646.53045
[10] On the moduli of vector bundles on an algebraic surface, Ann. of Math. (2), Tome 106 (1977) no. 1, pp. 45-60 | Article | MR 466475 | Zbl 0381.14003
[11] Analytic Hilbert quotients, Several complex variables (Berkeley, CA, 1995-1996), Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge, Tome 37 (1999), pp. 309-349 | MR 1748608 | Zbl 0959.32013
[12] Cartan decomposition of the moment map, Math. Ann., Tome 337 (2007) no. 1, pp. 197-232 | Article | MR 2262782 | Zbl 1110.32008
[13] Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematic, Academic Press Inc., XV. 628 p., New York Tome 80 (1978) | MR 514561 | Zbl 0451.53038
[14] Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Springer-Verlag, New York Tome 9 (1978) (Second printing, revised) | MR 499562 | Zbl 0447.17001
[15] The length of vectors in representation spaces, Algebraic geometry. (Proc. Summer Meeting, Copenhagen, 1978), Lecture Notes in Math., Springer, Berlin, Tome 732 (1979), pp. 233-243 | MR 555701 | Zbl 0407.22012
[16] Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, Princeton University Press, Princeton, NJ, Tome 15 (1987) (Kanô Memorial Lectures, 5) | MR 909698 | Zbl 0708.53002
[17] On filtered Lie algebras and geometric structures. II, J. Math. Mech., Tome 14 (1965), pp. 513-521 | MR 185042 | Zbl 0163.28103
[18] Sur les orbites fermées des groupes algébriques réductifs, Invent. Math., Tome 16 (1972), pp. 1-5 | Article | MR 294351 | Zbl 0249.14016
[19] Geometric invariant theory, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], Tome 34 (1994) (third edition) | MR 1304906 | Zbl 0797.14004
[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 1064110 | Zbl 0722.22004
[21] Spinor bundles on quadrics, Trans. Amer. Math. Soc., Tome 307 (1988) no. 1, pp. 301-316 | Article | MR 936818 | Zbl 0657.14006
[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, Tome 5 (1966), pp. 159-177 | Article | MR 190947 | Zbl 0138.18602
[24] On a theorem of Ramanan, Nagoya Math. J., Tome 69 (1978), pp. 131-138 | MR 473243 | Zbl 0345.14017
[25] Balance point and stability of vector bundles over a projective manifold, Math. Res. Lett., Tome 9(2-3) (2002), pp. 393-411 | Article | MR 1909652 | Zbl 1011.32016