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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

14F25 Classical real and complex (co)homology in algebraic geometry
14M17 Homogeneous spaces and generalizations
14L30 Group actions on varieties or schemes (quotients)
Full Text: DOI EuDML
[1] AKYILDIZ, E.: Bruhat decomposition via Gm-action, Bull. Acad. Pol. Sci., Sér. Sci. Math.28, 541-547 (1980) · Zbl 0483.20024
[2] AKYILDIZ, E.: Vector fields and equivariant bundles, Pac. Jour. of Math.,81, 283-289 (1979) · Zbl 0377.14004
[3] AKYILDIZ, E.: Vector fields and cohomology of G/P, Lecture Notes in Mathematics956, Springer-Verlag, 1-9 (1982)
[4] AKYILDIZ, E., CARRELL, J.B., LIEBERMAN, D.I.: Zeros of holomorphic vector fields on singular spaces and intersection rings of Schubert varieties, Compositio Math.57, 237-248 (1986) · Zbl 0613.14035
[5] AKYILDIZ, E., CARRELL, J.B., LIEBERMAN, D.I., SOMMESE, A.J.: On the graded rings associated to holomorphic vector fields with exactly one zero, Proc. Symp. Pure Math.40, 55-56 (1983) · Zbl 0523.57031
[6] BORHO, W., KRAFT, H.: Über Bahnen und deren Deformation bei linearen Aktionen reduktiver Gruppen, Comment. Math. Helv.54, 61-104 (1979) · Zbl 0395.14013
[7] CARRELL, J.B.: Vector fields and cohomology of G/B, Progress in Math.14, Birkhauser, 57-65 (1981)
[8] CARRELL, J.B., LIEBERMAN, D.I.: Holomorphic vector fields and compact Kaehler manifolds, Invent. Math.21, 303-309 (1973) · Zbl 0258.32013
[9] CARRELL, J.B., LIEBERMAN, D.I.: Vector fields and Chern numbers, Math. Ann.225, 263-273 (1977) · Zbl 0365.32020
[10] CARRELL, J.B., SOMMESE, A.J.: SL(2, C) actions on compact Kaehler manifolds, Tran. of Amer. Math. Soc.276, 165-179 (1983) · Zbl 0591.32034
[11] GRIFFITHS, P., HARRIS, H.: Principles of Algebraic Geometry, John Wiley and Sons, New York (1978) · Zbl 0408.14001
[12] KOSTANT, B.: The principal three-dimensional subgroup and the Betti numbers of complex semisimple Lie group, Amer. Jour. Math.81, 973-1032 (1959) · Zbl 0099.25603
[13] KOSTANT, B.: Lie group presentations on polynomial rings, Amer. Jour. Math.85, 327-404 (1963) · Zbl 0124.26802
[14] KOSTANT, B.: On Whittaker vectors and representation theory, Invent. Math.48, 101-184 (1978) · Zbl 0405.22013
[15] KRAFT, H.: Conjugacy classes and Weyl group representations, Astérisque 87-88, 191-205 (1981) · Zbl 0489.17002
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