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Zbl 1161.53064
Biliotti, Leonardo; Ghigi, Alessandro
Homogeneous bundles and the first eigenvalue of symmetric spaces. (English)
[J] Ann. Inst. Fourier 58, No. 7, 2315-2331 (2008). ISSN 0373-0956; ISSN 1777-5310/e

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 $H^0(E)\ne0$, then $T_E$ is stable. The authors give two proofs -- the first is algebraic and uses a criterion of {\it 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 {\it 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 $\lambda_1\le2$ 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 $\lambda_1\le 36/17$ resp. $\lambda_1\le 133/53$.
[Peter B. Gilkey (Eugene)]

MSC 2000:: *53C55 Complex differential geometry (global)
58J50 Spectral problems; spectral geometry; scattering theory
32M10 Homogeneous complex manifolds

Keywords: homogeneous bundle; spectrum of the Laplacian; Gieseker point; compact Hermitian symmetric space

Citations: Zbl 0249.14016; Zbl 1011.32016

Cited in: Zbl 1206.53078

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