# 2022-01-28

Tags: #untagged

Refs: ?

## 15:11

• Bott-Samelson manifold: lists of flags starting at the base flag $$B/B$$ only changing by a little bit at each step.
• It’s an iterated $${\mathbb{P}}^1$$ bundle, hence smooth projective irreducible with a $$B$$ action. $$\BS^Q$$ associated to a word $$Q$$ in the Weyl group.
• Bott-Samelson map $$\BS^Q \to G/B$$: send a flag to the top space, its image is some $$B{\hbox{-}}$$orbit closure $$X^w = { \operatorname{cl}} (BwB)/B$$, i.e. a Schubert variety. Yields that $$\BS^Q \to X^w$$ is a resolution of singularities, the advantage is that it’s combinatorially easy to understand. Rational, degree 1.
• What is a moment map?
• What is a stratification?
#web/quick_notes #untagged