zbMATH — the first resource for mathematics

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
Full Text:
References:
 [1] Serre, J.P.: Exposé au séminaire Bourbaki, no 100 (1954) [2] Brion, M.: Points entiers dans les polyèdres convexes. Ann. Sci. Ec. Norm. Super., IV. Ser.4, 21, 653-663 (1988) · Zbl 0667.52011 [3] Brion, M.: Groupe de Picard et nombres caractéristiques des variétés sphériques. Duke Math. J.58, 397-424 (1989) · Zbl 0701.14052 · doi:10.1215/S0012-7094-89-05818-3 [4] De Concini, C., Procesi, c.: Complete symmetric varieties. I. Invariant theory. Lect. Notes Math., vol. 996. Berlin Heidelberg New York: Springer 1983 · Zbl 0581.14041 [5] Kawamata, Y.: A generalization of Kodaira-Ramanujam’s vanishing theorem. Math. Ann.261, 43-46 (1982) · Zbl 0488.14003 · doi:10.1007/BF01456407 [6] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal embeddings I (Lect. Notes Math., vol. 339). Berlin Heidelberg New York: Springer 1983 · Zbl 0271.14017 [7] Kempf, G.: On the collapsing of homogeneous bundles. Invent. Math.37, 229-239 (1976) · Zbl 0338.14015 · doi:10.1007/BF01390321 [8] Oda, T.: Convex bodies and algebraic geometry (Ergeb. Math. Grenzgeb., vol. 15). Berlin Heidelberg New York: Springer 1988 · Zbl 0628.52002 [9] Popov, V.L.: Contraction of the actions of reductive algebraic groups. Math. USSR. Sb.58, 311-335 (1987) · Zbl 0627.14033 · doi:10.1070/SM1987v058n02ABEH003106 [10] Strickland, E.: A vanishing theorem for group compactifications. Math. Ann.277, 165-171 (1987) · Zbl 0595.14037 · doi:10.1007/BF01457285 [11] Sumihiro, H.: Equivariant completion. II. Math. Kyoto Univ.15, 573-605 (1975) · Zbl 0331.14008 [12] Viehweg, E.: Vanishing theorems. J. Crelle335, 1-8 (1982) · Zbl 0485.32019 · doi:10.1515/crll.1982.335.1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.