Stable bundles and the first eigenvalue of the Laplacian |
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Authors: | Claudio Arezzo Alessandro Ghigi Andrea Loi |
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Institution: | (1) Università di Parma, Italy;(2) Università di Milano Bicocca, Italy;(3) Università di Cagliari, Italy |
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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 Te is stable, then for any Kähler metric g on M\(\lambda _1 (M,g) \leqslant \frac{{4\pi h^0 (E)}}{{r(h^0 (E) - r)}} \cdot \frac{{\left\langle {C_1 (E) \cup \omega ]^{n - 1} ,M]} \right\rangle }}{{(n - 1)!vol(M,\omega ])}}\) where ω = ωg is the Kähler form associated to g.By this method we obtain, for example, a sharp upper bound for λ1 of Kähler metrics on complex Grassmannians. |
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Keywords: | Math Subject Classifications" target="_blank">Math Subject Classifications 58J50 53C55 14J60 |
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