Homogeneous bundles and the first eigenvalue of symmetric spaces

Leonardo Biliotti[1]; Alessandro Ghigi[2]

  • [1] Università degli Studi di Parma Parma (Italia)
  • [2] Università degli Studi di Milano Bicocca Milano (Italia)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 7, page 2315-2331
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Biliotti, Leonardo, and Ghigi, Alessandro. "Homogeneous bundles and the first eigenvalue of symmetric spaces." Annales de l’institut Fourier 58.7 (2008): 2315-2331. <http://eudml.org/doc/10379>.

@article{Biliotti2008,
abstract = {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.},
affiliation = {Università degli Studi di Parma Parma (Italia); Università degli Studi di Milano Bicocca Milano (Italia)},
author = {Biliotti, Leonardo, Ghigi, Alessandro},
journal = {Annales de l’institut Fourier},
keywords = {Homogeneous bundles; spectrum of the Laplacian; homogeneous bundle; Gieseker point; compact Hermitian symmetric space},
language = {eng},
number = {7},
pages = {2315-2331},
publisher = {Association des Annales de l’institut Fourier},
title = {Homogeneous bundles and the first eigenvalue of symmetric spaces},
url = {http://eudml.org/doc/10379},
volume = {58},
year = {2008},
}

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
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 7
SP - 2315
EP - 2331
AB - 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.
LA - eng
KW - Homogeneous bundles; spectrum of the Laplacian; homogeneous bundle; Gieseker point; compact Hermitian symmetric space
UR - http://eudml.org/doc/10379
ER -

References

top
  1. D. N. Akhiezer, Lie group actions in complex analysis, E27 (1995), Friedr. Vieweg & Sohn, Braunschweig Zbl0845.22001MR1334091
  2. C. Arezzo, A. Ghigi, A. Loi, Stable bundles and the first eigenvalue of the Laplacian, J. Geom. Anal. 17 (2007), 375-386 Zbl1128.58013MR2358762
  3. R. J. Baston, M. G. Eastwood, The Penrose transform, (1989), The Clarendon Press Oxford University Press, New York Zbl0726.58004MR1038279
  4. J.-P. Bourguignon, P. Li, S.-T. Yau, Upper bound for the first eigenvalue of algebraic submanifolds, Comment. Math. Helv. 69 (1994), 199-207 Zbl0814.53040MR1282367
  5. B. Colbois, J. Dodziuk, Riemannian metrics with large λ 1 , Proc. Amer. Math. Soc. 122 (1994), 905-906 Zbl0820.58056MR1213857
  6. S. K. Donaldson, P. B. Kronheimer, The geometry of four-manifolds, (1990), Oxford Mathematical Monographs. Oxford: Clarendon Press. ix, 440p., New York Zbl0820.57002MR1079726
  7. A. El Soufi, S. Ilias, Riemannian manifolds admitting isometric immersions by their first eigenfunctions, Pacific J. Math. 195 (2000), 91-99 Zbl1030.53043MR1781616
  8. G. Fels, A. Huckleberry, J. A. Wolf, Cycle spaces of flag domains, 245 (2006), Birkhäuser Boston Inc., Boston, MA Zbl1084.22011MR2188135
  9. A. Futaki, Kähler-Einstein metrics and integral invariants, (1988), Springer-Verlag, Berlin Zbl0646.53045MR947341
  10. D. Gieseker, On the moduli of vector bundles on an algebraic surface, Ann. of Math. (2) 106 (1977), 45-60 Zbl0381.14003MR466475
  11. P. Heinzner, A. Huckleberry, Analytic Hilbert quotients, Several complex variables (Berkeley, CA, 1995-1996) 37 (1999), 309-349, Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge Zbl0959.32013MR1748608
  12. P. Heinzner, G. W. Schwarz, Cartan decomposition of the moment map, Math. Ann. 337 (2007), 197-232 Zbl1110.32008MR2262782
  13. S. Helgason, Differential geometry, Lie groups, and symmetric spaces, 80 (1978), Pure and Applied Mathematic, Academic Press Inc., XV. 628 p., New York Zbl0451.53038MR514561
  14. J. E. Humphreys, Introduction to Lie algebras and representation theory, 9 (1978), Graduate Texts in Mathematics, Springer-Verlag, New York Zbl0447.17001MR499562
  15. G. Kempf, L. Ness, The length of vectors in representation spaces, Algebraic geometry. (Proc. Summer Meeting, Copenhagen, 1978) 732 (1979), 233-243, Lecture Notes in Math., Springer, Berlin Zbl0407.22012MR555701
  16. S. Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan 15 (1987), Princeton University Press, Princeton, NJ Zbl0708.53002MR909698
  17. S. Kobayashi, T. Nagano, On filtered Lie algebras and geometric structures. II, J. Math. Mech. 14 (1965), 513-521 Zbl0163.28103MR185042
  18. D. Luna, Sur les orbites fermées des groupes algébriques réductifs, Invent. Math. 16 (1972), 1-5 Zbl0249.14016MR294351
  19. D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory, 34 (1994), Springer-Verlag, Berlin Zbl0797.14004MR1304906
  20. A. L. Onishchik, È. B. Vinberg, Lie groups and algebraic groups, (1990), Springer Series in Soviet Mathematics, Springer-Verlag, Berlin Zbl0722.22004MR1064110
  21. G. Ottaviani, Spinor bundles on quadrics, Trans. Amer. Math. Soc. 307 (1988), 301-316 Zbl0657.14006MR936818
  22. G. Ottaviani, Rational homogeneous varieties, Notes from a course held in Cortona, Italy (1995) 
  23. S. Ramanan, Holomorphic vector bundles on homogeneous spaces, Topology 5 (1966), 159-177 Zbl0138.18602MR190947
  24. H. Umemura, On a theorem of Ramanan, Nagoya Math. J. 69 (1978), 131-138 Zbl0345.14017MR473243
  25. X. Wang, Balance point and stability of vector bundles over a projective manifold, Math. Res. Lett. 9(2-3) (2002), 393-411 Zbl1011.32016MR1909652

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.