Bott–Samelson resolution

Let G be a connected reductive complex algebraic group, B a Borel subgroup and T a maximal torus contained in B.

Let $w\; \backslash in\; W\; =\; N\_G(T)/T.$ Any such w can be written as a product of reflections by simple roots. Fix minimal such an expression:

:$\backslash underline\{w\}\; =\; (s\_\{i\_1\},\; s\_\{i\_2\},\; \backslash ldots,\; s\_\{i\_\backslash ell\})$

so that $w\; =\; s\_\{i\_1\}\; s\_\{i\_2\}\; \backslash cdots\; s\_\{i\_\backslash ell\}$. (ℓ is the length of w.) Let $P\_\{i\_j\}\; \backslash subset\; G$ be the subgroup generated by B and a representative of $s\_\{i\_j\}$. Let $Z\_\{\backslash underline\{w\}\}$ be the quotient:

:$Z\_\{\backslash underline\{w\}\}\; =\; P\_\{i\_1\}\; \backslash times\; \backslash cdots\; \backslash times\; P\_\{i\_\backslash ell\}/B^\backslash ell$

with respect to the action of $B^\backslash ell$ by

:$(b\_1,\; \backslash ldots,\; b\_\backslash ell)\; \backslash cdot\; (p\_1,\; \backslash ldots,\; p\_\backslash ell)\; =\; (p\_1\; b\_1^\{-1\},\; b\_1\; p\_2\; b\_2^\{-1\},\; \backslash ldots,\; b\_\{\backslash ell-1\}\; p\_\backslash ell\; b\_\backslash ell^\{-1\}).$

It is a smooth projective variety. Writing $X\_w\; =\; \backslash overline\{BwB\}\; /\; B\; =\; (P\_\{i\_1\}\; \backslash cdots\; P\_\{i\_\backslash ell\})/B$ for the Schubert variety for w, the multiplication map

:$\backslash pi:\; Z\_\{\backslash underline\{w\}\}\; \backslash to\; X\_w$

is a resolution of singularities called the Bott–Samelson resolution. $\backslash pi$ has the property: $\backslash pi\_*\; \backslash mathcal\{O\}\_\{Z\_\{\backslash underline\{w\}\}\}\; =\; \backslash mathcal\{O\}\_\{X\_w\}$ and $R^i\; \backslash pi\_*\; \backslash mathcal\{O\}\_\{Z\_\{\backslash underline\{w\}\}\}\; =\; 0,\; \backslash ,\; i\; \backslash ge\; 1.$ In other words, $X\_w$ has rational singularities.{{harvtxt|Brion|2005|loc=Theorem 2.2.3.}}

There are also some other constructions; see, for example, {{harvtxt|Vakil|2006}}.

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{{DEFAULTSORT:Bott-Samelson resolution}}

Category:Algebraic geometry

Category:Singularity theory