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Cohomology of projective varieties with regular $$SL_ 2$$ actions. (English) Zbl 0626.14017
Authors’ abstract: Let G be a complex semisimple linear algebraic group, B a fixed Borel subgroup of G, H a maximal torus of G in B, $${\mathfrak g}$$ and $${\mathfrak h}$$ the Lie algebras of G and H, respectively. Kostant has expressed the cohomology ring of G/B as the coordinate ring A($${\mathfrak N}\cap {\mathfrak h})$$ of the scheme theoretic intersection $${\mathfrak N}\cap {\mathfrak h}$$ of the variety of nilpotent elements $${\mathfrak N}$$ of $${\mathfrak g}$$ with $${\mathfrak h}$$. The purpose of this paper is to give a similar description of the cohomology ring of a nonsingular complex projective variety X with a “regular” $$SL_ 2$$ action. We will show that there is an intrinsically defined subscheme Z of X whose coordinate ring A(Z) is isomorphic to the cohomology ring of X. When $$X=G/B$$, we will identify A(Z) with Kostant’s description A($${\mathfrak N}\cap {\mathfrak h})$$.
Reviewer: F.Pauer

##### MSC:
 14F25 Classical real and complex (co)homology in algebraic geometry 14M17 Homogeneous spaces and generalizations 14L30 Group actions on varieties or schemes (quotients)
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