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Une extension du théorème de Borel-Weil. (An extension of the Borel- Weil theorem). (French) Zbl 0718.14011
Fix a reductive group. A spherical variety is one on which this group acts so that some Borel subgroup has a dense orbit. The main result shows that for any proper equivariant morphism \(\pi\) of spherical varieties the higher derived functors \(R^ i\pi_*L (i>0)\) vanish for any line bundle L generated by its global sections.
The proof is by reduction to the case of a birational morphism with rational singularities and L trivial. Consequences are that \(L^{-1}\) has only one non-vanishing cohomology group, corresponding to the Kodaira dimension of L, and a generalisation replacing L by an integrally closed fractional ideal sheaf.

MSC:
14F17 Vanishing theorems in algebraic geometry
14L30 Group actions on varieties or schemes (quotients)
14C20 Divisors, linear systems, invertible sheaves
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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