Bott–Samelson resolution

In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities. It was introduced by Template:Harvtxt in the context of compact Lie groups.Template:Sfnp The algebraic formulation is independently due to Template:Harvtxt and Template:Harvtxt.

Definition

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

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

$\underset{\_}{w}=(s_{i_1},s_{i_2},\dots ,s_{i_{\ell }})$

so that $w=s_{i_1}{s}_{i_2}\cdots s_{i_{\ell }}$. (ℓ is the length of w.) Let $P_{i_j}\subset G$ be the subgroup generated by B and a representative of $s_{i_j}$. Let $Z_{\underset{\_}{w}}$ be the quotient:

$Z_{\underset{\_}{w}}=P_{i_1}\times \cdots \times P_{i_{\ell }}/B^{\ell }$

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

$(b_1,\dots ,b_{\ell })\cdot (p_1,\dots ,p_{\ell })=(p_1{b}_1^{-1},b_1{p}_2{b}_2^{-1},\dots ,b_{\ell -1}{p}_{\ell }{b}_{\ell }^{-1}).$

It is a smooth projective variety. Writing $X_w=\overline{BwB}/B=(P_{i_1}\cdots P_{i_{\ell }})/B$ for the Schubert variety for w, the multiplication map

$\pi :Z_{\underset{\_}{w}}\to X_w$

is a resolution of singularities called the Bott–Samelson resolution. $\pi$ has the property: ${\pi }_{*}{𝒪}_{Z_{\underset{\_}{w}}}={𝒪}_{X_w}$ and $R^i{\pi }_{*}{𝒪}_{Z_{\underset{\_}{w}}}=0,i\ge 1.$ In other words, $X_w$ has rational singularities.

There are also some other constructions; see, for example, Template:Harvtxt.

References

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