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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 \({\mathbf{P}}^1\) 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?