A vanishing theorem for open orbits on complex flag manifolds
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- by Wilfried Schmid and Joseph A. Wolf PDF
- Proc. Amer. Math. Soc. 92 (1984), 461-464 Request permission
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.References
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Additional Information
- © Copyright 1984 American Mathematical Society
- 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