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A Borel-Weil theorem for holomorphic forms. (English) Zbl 0859.22004

This paper proves vanishing theorems of Dolbeault cohomology for homogeneous vector bundles with dominant weights on homogeneous spaces of semisimple complex Lie groups. Let \(G\) be a semisimple complex Lie group, \(P\) be a parabolic subgroup of \(G\), and \(E\) be a homogeneous vector bundle on \(X=G/P\) whose associated highest \(P\)-weights are dominant. A consequence of Bott’s theorem implies that \(H^q(X,E)=0\) for \(q>0\). This paper extends this result to Dolbeault cohomology, \(H^{p,q}(X,E)=H^q(X,\Omega^p_X \otimes E)\), where \(\Omega^p_X\) is the bundle of holomorphic \(p\)-forms on \(X\).
The main theorem is that \(H^{p,q}(X,E) = 0\) whenever \(q>2p\) and \(G\) is a product of simply connected semisimple Lie groups with no factor of type \(E_7\) or \(E_8\). An important technique is the reduction to symmetric space towers: \(X\) is called a symmetric space tower if there exist fibrations \(X \to X_1 \to \cdots \to Y_s \to \{0\}\) whose fibers are compact Hermitian symmetric spaces. For such a space, the authors prove that \(H^{p,q}(X,E)=0\) whenever \(q>p\) (in fact this is always sharp). As a corollary, they obtain refined versions of the Nakano vanishing theorem for ample line bundles.

MSC:

22E10 General properties and structure of complex Lie groups
22E46 Semisimple Lie groups and their representations
32M05 Complex Lie groups, group actions on complex spaces
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References:

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