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Stable bundles and the first eigenvalue of the Laplacian. (English) Zbl 1128.58013
From the authors’ abstract: In this article we study the first eigenvalue of the Laplacian on a compact manifold using stable bundles and balanced bases. Our main result is the following: Let \(M\) be a compact Kähler manifold of complex dimension \(n\) and \(E\) a holomorphic vector bundle of rank \(r\) over \(M\). If \(E\) is globally generated and its Gieseker point \(T_{E}\) is stable, then for any Kähler metric \(g\) on \(M\) \[ \lambda_{1}(M,g)\leq\frac{4\pi h^{0}(E)}{r(h^{0}(E)-r)}\cdot \frac{\langle c_{1}(E)\cup[\omega]^{n-1},[M]\rangle}{(n-1)!\text{vol}(M,[\omega])}, \] where \(\omega=\omega_{g}\) is the Kähler metric associated to \(g\). By this method we obtain, for example, a sharp upper bound for \(\lambda_ {1}\) of Kähler metrics on complex Grassmannians.

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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