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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
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