Bott–Samelson resolution

In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities. It was introduced by Bott & Samelson (1958) in the context of compact Lie groups.[1] The algebraic formulation is independently due to Hansen (1973) and Demazure (1974).

Definition

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

Let   Any such w can be written as a product of reflections by simple roots. Fix minimal such an expression:

 

so that  . ( is the length of w.) Let   be the subgroup generated by B and a representative of  . Let   be the quotient:

 

with respect to the action of   by

 

It is a smooth projective variety. Writing   for the Schubert variety for w, the multiplication map

 

is a resolution of singularities called the Bott–Samelson resolution.   has the property:   and   In other words,   has rational singularities.[2]

There are also some other constructions; see, for example, Vakil (2006).

Notes

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References

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  • Bott, Raoul; Samelson, Hans (1958), "Applications of the theory of Morse to symmetric spaces", American Journal of Mathematics, 80: 964–1029, doi:10.2307/2372843, MR 0105694.
  • Brion, Michel (2005), "Lectures on the geometry of flag varieties", Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, pp. 33–85, arXiv:math/0410240, doi:10.1007/3-7643-7342-3_2, MR 2143072.
  • Demazure, Michel (1974), "Désingularisation des variétés de Schubert généralisées", Annales Scientifiques de l'École Normale Supérieure (in French), 7: 53–88, MR 0354697.
  • Gorodski, Claudio; Thorbergsson, Gudlaugur (2002), "Cycles of Bott-Samelson type for taut representations", Annals of Global Analysis and Geometry, 21 (3): 287–302, arXiv:math/0101209, doi:10.1023/A:1014911422026, MR 1896478.
  • Hansen, H. C. (1973), "On cycles in flag manifolds", Mathematica Scandinavica, 33: 269–274 (1974), doi:10.7146/math.scand.a-11489, MR 0376703.
  • Vakil, Ravi (2006), "A geometric Littlewood-Richardson rule", Annals of Mathematics, Second Series, 164 (2): 371–421, arXiv:math.AG/0302294, doi:10.4007/annals.2006.164.371, MR 2247964.