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Positivity and vanishing theorems for ample vector bundles. (English) Zbl 1264.14057

There are several notions of positivity for vector bundles (Nakano, Griffiths, Hartshorne), and some points concerning their interplay are not yet completely clarified. In the paper under review, the authors study the Nakano-positivity and the dual Nakano-positivity for certain vector bundles naturally associated with an ample vector bundle. For example, for an ample vector bundle \(E\) over a compact Kähler manifold \(X\) of dimension \(n\), they prove that \(S^kE \otimes \det E\) is both Nakano-positive and dual Nakano-positive for any integer \(k \geq 0\). Moreover, applying this conclusion to suitable adjoint bundles, they obtain new vanishing theorems, for instance the vanishing of \(H^{n,q}(X, S^kE \otimes \det E)\) and \(H^{q,n}(X, S^kE \otimes \det E)\) for any \(q \geq 1\), \(k \geq 0\) and any ample vector bundle \(E\). Finally, comparing the Griffiths-positivity with the Nakano-positivity, the authors prove the Nakano-positivity and the dual Nakano-positivity of the Hermitian vector bundle \(S^kE \otimes \det E\) associated with a Griffiths-positive vector bundle \(E\).

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
32J27 Compact Kähler manifolds: generalizations, classification
32L20 Vanishing theorems
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