Frobenius Splitting Methods in Geometry and Representation Theory

Michel Brion and Shrawan Kumar

Contents

1. Frobenius splitting: general theory

1. Basic definitions, properties, and examples
2. Consequences of Frobenius splitting
3. Criteria for splitting
4. Splitting relative to a divisor
5. Consequences of diagonal splitting
6. From characteristic p to characteristic 0

2. Frobenius splitting of Schubert varieties

1. Notation
2. Frobenius splitting of BSDH varieties Z_w
3. Some more splittings of G/B and G/B x G/B

3. Cohomology and geometry of Schubert varieties

1. Cohomology of Schubert varieties
2. Normality of Schubert varieties
3. Demazure character formula
4. Schubert varieties have rational resolutions
5. Homogeneous coordinate rings of Schubert varieties are Koszul algebras

4. Canonical splitting and good filtration

1. Canonical splitting
2. Good filtrations
3. Proof of the PRVK conjecture and its refinement

5. Cotangent bundles of flag varieties

1. Splitting of cotangent bundles of flag varieties
2. Cohomology vanishing of cotangent bundles of flag varieties
3. Geometry of the nilpotent and subregular cones

6. Equivariant embeddings of reductive groups

1. The wonderful compactification
2. Reductive embeddings

7. Hilbert schemes of points on surfaces

1. Symmetric products
2. Hilbert schemes of points
3. The Hilbert-Chow morphism
4. Hilbert schemes of points on surfaces
5. Splitting of Hilbert schemes of points on surfaces