Book contents
- Frontmatter
- Contents
- Preface
- 1 Introductory Material
- 2 Schur Functors and Schur Complexes
- 3 Grassmannians and Flag Varieties
- 4 Bott's Theorem
- 5 The Geometric Technique
- 6 The Determinantal Varieties
- 7 Higher Rank Varieties
- 8 The Nilpotent Orbit Closures
- 9 Resultants and Discriminants
- References
- Notation Index
- Subject Index
4 - Bott's Theorem
Published online by Cambridge University Press: 18 August 2009
- Frontmatter
- Contents
- Preface
- 1 Introductory Material
- 2 Schur Functors and Schur Complexes
- 3 Grassmannians and Flag Varieties
- 4 Bott's Theorem
- 5 The Geometric Technique
- 6 The Determinantal Varieties
- 7 Higher Rank Varieties
- 8 The Nilpotent Orbit Closures
- 9 Resultants and Discriminants
- References
- Notation Index
- Subject Index
Summary
This chapter is devoted to the discussion of several versions of Bott's theorem on cohomology of line bundles on homogeneous spaces G/B. We also prove some corresponding results on cohomology of some vector bundles on homogeneous spaces G/P. They follow by applying the spectral sequence of the composition to a line bundle on G/B.
Throughout the chapter we prove all results for general linear groups and state them for arbitrary reductive groups. This allows us to avoid technicalities of general theory while preserving the basic ideas of the proof.
In section 4.1 we classify line bundles on flag varieties. They correspond to integral weights for the general linear group. We also state Bott's theorem for the general linear groups. We prove related results on cohomology of some vector bundles on Grassmannians and partial flag varieties. Finally we state Kempf's vanishing theorem.
In section 4.2 we prove Bott's theorem for the general linear group. The proof follows closely the approach of Demazure [D].
In section 4.3 we state Bott's theorem for an arbitrary reductive group. We also give some explicit calculations of cohomology on homogeneous spaces G/P where G is a classical group and P is a maximal parabolic subgroup. They will be needed in chapter 8.
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- Chapter
- Information
- Cohomology of Vector Bundles and Syzygies , pp. 110 - 135Publisher: Cambridge University PressPrint publication year: 2003