Fixed points for reductive group actions on acyclic varieties
Annales de l'Institut Fourier, Tome 45 (1995) no. 5, pp. 1249-1281.

Soit X une variété complexe, affine et lisse, qui, considérée comme variété analytique, a la -cohomologie singulière d’un point. Supposons que G soit un groupe complexe algébrique agissant algébriquement sur X. Nos résultats principaux sont les suivants : Si G est semisimple, la fibre générique de l’application quotient π:XX//G contient une orbite dense. Si G est connexe et réductif, l’action a des points fixes si dim X//G3.

Let X be a smooth, affine complex variety, which, considered as a complex manifold, has the singular -cohomology of a point. Suppose that G is a complex algebraic group acting algebraically on X. Our main results are the following: if G is semi-simple, then the generic fiber of the quotient map π:XX//G contains a dense orbit. If G is connected and reductive, then the action has fixed points if dim X//G3.

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     author = {Fankhauser, Martin},
     title = {Fixed points for reductive group actions on acyclic varieties},
     journal = {Annales de l'Institut Fourier},
     pages = {1249--1281},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {5},
     year = {1995},
     doi = {10.5802/aif.1495},
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     mrnumber = {97a:14047},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1495/}
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Fankhauser, Martin. Fixed points for reductive group actions on acyclic varieties. Annales de l'Institut Fourier, Tome 45 (1995) no. 5, pp. 1249-1281. doi : 10.5802/aif.1495. https://aif.centre-mersenne.org/articles/10.5802/aif.1495/

[Ba] H. Bass, A non-triangular action of Ga on A3, Jour. Pure Appl. Alg., 33 (1984), 1-5. | MR | Zbl

[BdS] A. Borel and J. De Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helvetici, 23 (1949), 200-221. | MR | Zbl

[Bo] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968.

[Br] G.E. Bredon, Introduction to compact transformation groups, pure and applied mathematics, Volume 46, Academic Press, New York and London, 1972. | MR | Zbl

[El] A.G. Elashvili, Canonical form and stationary subalgebras of points of general position for simple linear groups, Functional Analysis and its Applications, 6 (1972), 44-53. | MR | Zbl

[Fa] M. Fankhauser, Reductive Group Actions on Acyclic Varieties, Thesis, Basel, 1994.

[Hs] W.-Y. Hsiang, On the Geometric Weight System of Differentiable Compact Transformation Groups on Acyclic Manifolds, Inventiones Math., 12 (1971), 35-47. | MR | Zbl

[HH70] W.-C. Hsiang and W.-Y. Hsiang, Differentiable actions of compact connected classical groups : II, Annals of Mathematics, 92 (1970), 189-223. | MR | Zbl

[HH74] W.-C. Hsiang and W.-Y. Hsiang, Differentiable actions of compact connected Lie groups : III, Annals of Mathematics, 99 (1974), 220-256. | MR | Zbl

[HS82] W.-Y. Hsiang and E. Straume, Actions of compact connected Lie Groups with few orbit types, J. Reine Angew. Math., 334 (1982), 1-26. | MR | Zbl

[HS86] W.-Y. Hsiang and E. Straume, Actions of compact connected Lie groups on acyclic manifolds with low dimensional orbit spaces, J. Reine Angew. Math., 369 (1986), 21-39. | MR | Zbl

[Hu] J. E. Humphreys, Linear Algebraic Groups, GTM 21, Springer-Verlag, New York-Heidelberg-Berlin, 1987.

[Kr84] H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspekte der Mathematik, band D1, Vieweg, Braunschweig, 1984. | MR | Zbl

[Kr89a] H. Kraft, G-vector bundles and the linearization problem, Contemporary Mathematics, 10 (1989), 111-123. | MR | Zbl

[Kr89b] H. Kraft, Algebraic automorphisms of affine space in : Topological Methods in Algebraic Transformation Groups, Progress in Mathematics, Volume 80, Birkhäuser-Verlag, Boston-Basel-Berlin, 1989, pp. 81-106. | MR | Zbl

[KP] H. Kraft and V.L. Popov, Semisimple group actions on three dimensional affine space are linear, Comment. Math. Helvetici, 60 (1985), 466-479. | MR | Zbl

[KS] H. Kraft and G. Schwarz, Reductive group actions with one-dimensional quotient, Publications Mathématiques IHES, 76 (1992), 1-97. | Numdam | MR | Zbl

[Lu] D. Luna, Slices étales, Bull. Soc. Math. France, Mémoire 33 (1973), 81-105. | Numdam | MR | Zbl

[Ol] R. Oliver, Weight systems for SO(3)-actions, Annals of Mathematics, 110 (1979), 227-241. | MR | Zbl

[PR] T. Petrie and J.D. Randall, Finite-order algebraic automorphisms of affine varieties, Comment. Math. Helvetici, 61 (1986), 203-221. | MR | Zbl

[SK] M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., 65 (1977), 1-155. | MR | Zbl

[Sch] G.W. Schwarz, Exotic algebraic group actions, C R. Acad. Sci. Paris, 309 (1989), 89-94. | MR | Zbl

[Ve] J.-L. Verdier, Caractéristique d'Euler-Poincaré, Bull. Soc. Math. France, 176 (1973), 441-445. | Numdam | MR | Zbl

[Vi] È.B. Vinberg, The Weyl Group of a graded Lie Algebra, Izv. Akad. Nauk SSSR, 40 (1976), 463-495. | Zbl

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