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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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