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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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