Borel-Weil-Bott theorem and geometry of Schubert varieties

We take the base field to be the field of complex numbers in these lectures. The varieties are, by definition, quasi-projective, reduced (but not necessarily irreducible) schemes. Let G be a semisimple, simply-connected, complex algebraic group with a fixed Borel subgroup B, a maximal torus H ⊂ B, and associated Weyl group W . (Recall that a Borel subgroup is any maximal connected, solvable subgroup; any two of which are conjugate to each other.) For any w ∈ W , we have the Schubert variety Xw := BwB/B ⊂ G/B. Also, let X(H) be the group of characters of H and X(H)+ the semigroup of dominant characters. For any λ ∈ X(H), we have the homogeneous line bundle L(λ) on G/B (cf. Section 5) and its restriction (denoted by the same symbol) to any Xw. The Lie algebras of G, B, andH are given by g, b, and h, respectively. For a fixed B, any subgroup P ⊂ G containing B is called a standard parabolic. The aim of these talks is to prove the following well-known results on the geometry and cohomology of Schubert varieties. Extension of these results to a connected reductive group is fairly straight forward. (1) Borel-Weil theorem and its generalization to the Borel-Weil-Bott theorem. (2) Any Schubert variety Xw is normal, and has rational singularities (in particular, is Cohen-Macaulay). (3) For any λ ∈ X(H)+, the linear system on Xw given by L(λ+ρ) embeds Xw as a projectively normal and projectively Cohen-Macaulay variety, where ρ is the half sum of positive roots.