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Proceedings of the American Mathematical Society

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A vanishing theorem for open orbits on complex flag manifolds


Authors: Wilfried Schmid and Joseph A. Wolf
Journal: Proc. Amer. Math. Soc. 92 (1984), 461-464
MSC: Primary 32F10; Secondary 22E46, 32L20
DOI: https://doi.org/10.1090/S0002-9939-1984-0759674-9
MathSciNet review: 759674
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Abstract: A real reductive Lie group $ G$ acts on complex flag manifolds $ {G_{\mathbf{C}}}$/(parabolic subgroup). The open orbits $ D = G(x)$ are precisely the homogeneous complex manifolds $ G/H$, where $ H$ is the centralizer of a torus. We prove that $ D$ is ($ (s + 1)$)-complete in the sense of Andreotti and Grauert, with $ s$ = complex dimension of a maximal compact subvariety of $ D$. Thus $ {H^q}(D,\mathcal{F}) = 0$ for $ q > s$ and any coherent sheaf $ \mathcal{F} \to D$. This vanishing theorem is needed for the realization of certain unitary representations on Dolbeault cohomology groups of homogeneous vector bundles.


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DOI: https://doi.org/10.1090/S0002-9939-1984-0759674-9
Article copyright: © Copyright 1984 American Mathematical Society

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