×

Homogeneous bundles and the first eigenvalue of symmetric spaces. (English) Zbl 1161.53064

The authors consider the Gieseker point of a homogeneous bundle over a rational homogeneous space and show:
Theorem 1.1: Let \(E\rightarrow X\) be an irreducible homogeneous vector bundle over a rational homogeneous space \(X=G/P\). If \(H^0(E)\neq0\), then \(T_E\) is stable.
The authors give two proofs – the first is algebraic and uses a criterion of D. Luna [Invent. Math. 16, 1–5 (1972; Zbl 0249.14016)] for an orbit to be closed. The second proof uses invariant metrics and uses a result of X. Wang [Math. Res. Lett. 9, No. 2–3, 393–411 (2002; Zbl 1011.32016)]. Theorem 1.1 is applied to the following problem in Kähler geometry. Let \(\lambda_1\) be the first eigenvalue of the Laplacian. The authors show:
Theorem 1.2: Let \(X\) be a compact irreducible Hermitian symmetric space of ABCD tpe. Then \(\lambda_1\leq2\) for any Kähler metric whose associated Kähler class lies in \(2\pi c_1(X)\). This bound is attained by the symmetric metric.
In the two exceptional examples of E type, the best estimate gotten by this method is strictly larger than 2 and is \(\lambda_1\) of the symmetric metric:
Theorem 1.3: If \(X=E_6/P(\alpha_1)\) resp. \(X=E_7/P(\alpha_7)\) then \(\lambda_1\leq 36/17\) resp. \(\lambda_1\leq 133/53\).

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
32M10 Homogeneous complex manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv Numdam EuDML

References:

[1] Akhiezer, D. N., Lie group actions in complex analysis, E27 (1995) · Zbl 0845.22001
[2] Arezzo, C.; Ghigi, A.; Loi, A., Stable bundles and the first eigenvalue of the Laplacian, J. Geom. Anal., 17, 3, 375-386 (2007) · Zbl 1128.58013
[3] Baston, R. J.; Eastwood, M. G., The Penrose transform (1989) · Zbl 0726.58004
[4] Bourguignon, J.-P.; Li, P.; Yau, S.-T., Upper bound for the first eigenvalue of algebraic submanifolds, Comment. Math. Helv., 69, 2, 199-207 (1994) · Zbl 0814.53040 · doi:10.1007/BF02564482
[5] Colbois, B.; Dodziuk, J., Riemannian metrics with large \(\lambda_1\), Proc. Amer. Math. Soc., 122, 3, 905-906 (1994) · Zbl 0820.58056
[6] Donaldson, S. K.; Kronheimer, P. B., The geometry of four-manifolds (1990) · Zbl 0820.57002
[7] El Soufi, A.; Ilias, S., Riemannian manifolds admitting isometric immersions by their first eigenfunctions, Pacific J. Math., 195, 1, 91-99 (2000) · Zbl 1030.53043 · doi:10.2140/pjm.2000.195.91
[8] Fels, G.; Huckleberry, A.; Wolf, J. A., Cycle spaces of flag domains, 245 (2006) · Zbl 1084.22011
[9] Futaki, A., Kähler-Einstein metrics and integral invariants (1988) · Zbl 0646.53045
[10] Gieseker, D., On the moduli of vector bundles on an algebraic surface, Ann. of Math. (2), 106, 1, 45-60 (1977) · Zbl 0381.14003 · doi:10.2307/1971157
[11] Heinzner, P.; Huckleberry, A., Several complex variables (Berkeley, CA, 1995-1996), 37, 309-349 (1999) · Zbl 0959.32013
[12] Heinzner, P.; Schwarz, G. W., Cartan decomposition of the moment map, Math. Ann., 337, 1, 197-232 (2007) · Zbl 1110.32008 · doi:10.1007/s00208-006-0032-8
[13] Helgason, S., Differential geometry, Lie groups, and symmetric spaces, 80 (1978) · Zbl 0451.53038
[14] Humphreys, J. E., Introduction to Lie algebras and representation theory, 9 (1978) · Zbl 0447.17001
[15] Kempf, G.; Ness, L., Algebraic geometry. (Proc. Summer Meeting, Copenhagen, 1978), 732, 233-243 (1979) · Zbl 0407.22012
[16] Kobayashi, S., Publications of the Mathematical Society of Japan, 15 (1987) · Zbl 0708.53002
[17] Kobayashi, S.; Nagano, T., On filtered Lie algebras and geometric structures. II, J. Math. Mech., 14, 513-521 (1965) · Zbl 0163.28103
[18] Luna, D., Sur les orbites fermées des groupes algébriques réductifs, Invent. Math., 16, 1-5 (1972) · Zbl 0249.14016 · doi:10.1007/BF01391210
[19] Mumford, D.; Fogarty, J.; Kirwan, F., Geometric invariant theory, 34 (1994) · Zbl 0797.14004
[20] Onishchik, A. L.; Vinberg, È. B., Lie groups and algebraic groups (1990) · Zbl 0722.22004
[21] Ottaviani, G., Spinor bundles on quadrics, Trans. Amer. Math. Soc., 307, 1, 301-316 (1988) · Zbl 0657.14006 · doi:10.1090/S0002-9947-1988-0936818-5
[22] Ottaviani, G., Rational homogeneous varieties, Notes from a course held in Cortona, Italy (1995)
[23] Ramanan, S., Holomorphic vector bundles on homogeneous spaces, Topology, 5, 159-177 (1966) · Zbl 0138.18602 · doi:10.1016/0040-9383(66)90017-6
[24] Umemura, H., On a theorem of Ramanan, Nagoya Math. J., 69, 131-138 (1978) · Zbl 0345.14017
[25] Wang, X., Balance point and stability of vector bundles over a projective manifold, Math. Res. Lett., 9(2-3), 393-411 (2002) · Zbl 1011.32016 · doi:10.1016/S0167-577X(02)00497-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.