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Line bundles on flag manifolds. (English) Zbl 0531.20022
Astérisque 87-88, 21-42 (1981).
In the paper under review the author gives a survey of some recent results on cohomology of line bundles on flag manifolds. It covers mostly the author’s own contributions to the theory but it also includes a brief account of the history of the subject. In the last section he mentions some of the other works that have been done in the area. A couple of open problems are also briefly discussed.
The following sections are in the paper. 1. The problem. 2. A little history. 3. $$H^ 1$$ [cf. H. H. Andersen, Invent. Math. 51, 287-296 (1979; Zbl 0417.20038)]. 4. The linkage principle [cf. H. H. Andersen, J. Reine Angew. Math. 315, 53-59 (1980; Zbl 0439.20026)]. 5. Further vanishing results. 6. An isomorphism theorem. 7. Related results and some open problems. The list of references includes 45 items.
For the entire collection see [Zbl 0468.00006].
Reviewer: N.I.Osetinski
##### MSC:
 20G10 Cohomology theory for linear algebraic groups 14M15 Grassmannians, Schubert varieties, flag manifolds 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 20G15 Linear algebraic groups over arbitrary fields 20G05 Representation theory for linear algebraic groups 14L35 Classical groups (algebro-geometric aspects)
##### Keywords:
survey; cohomology of line bundles; flag manifolds; problems