Fixed points for reductive group actions on acyclic varieties

Martin Fankhauser

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 5, page 1249-1281
  • ISSN: 0373-0956

Abstract

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Let X be a smooth, affine complex variety, which, considered as a complex manifold, has the singular ℤ -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 : X → X / / G contains a dense orbit. If G is connected and reductive, then the action has fixed points if dim X / / G ≤ 3 .

How to cite

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Fankhauser, Martin. "Fixed points for reductive group actions on acyclic varieties." Annales de l'institut Fourier 45.5 (1995): 1249-1281. <http://eudml.org/doc/75159>.

@article{Fankhauser1995,
abstract = {Let $X$ be a smooth, affine complex variety, which, considered as a complex manifold, has the singular $\{\Bbb Z\}$-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 $\pi :X\rightarrow X/\!\!/G$ contains a dense orbit. If $G$ is connected and reductive, then the action has fixed points if $\{\rm dim\}\,X/\!\!/G\le 3$.},
author = {Fankhauser, Martin},
journal = {Annales de l'institut Fourier},
keywords = {action of complex algebraic group; quotient; fixed points},
language = {eng},
number = {5},
pages = {1249-1281},
publisher = {Association des Annales de l'Institut Fourier},
title = {Fixed points for reductive group actions on acyclic varieties},
url = {http://eudml.org/doc/75159},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Fankhauser, Martin
TI - Fixed points for reductive group actions on acyclic varieties
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 5
SP - 1249
EP - 1281
AB - Let $X$ be a smooth, affine complex variety, which, considered as a complex manifold, has the singular ${\Bbb Z}$-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 $\pi :X\rightarrow X/\!\!/G$ contains a dense orbit. If $G$ is connected and reductive, then the action has fixed points if ${\rm dim}\,X/\!\!/G\le 3$.
LA - eng
KW - action of complex algebraic group; quotient; fixed points
UR - http://eudml.org/doc/75159
ER -

References

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  1. [Ba] H. BASS, A non-triangular action of Ga on A3, Jour. Pure Appl. Alg., 33 (1984), 1-5. Zbl0555.14019MR85j:14086
  2. [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. Zbl0034.30701MR11,326d
  3. [Bo] N. BOURBAKI, Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968. 
  4. [Br] G.E. BREDON, Introduction to compact transformation groups, pure and applied mathematics, Volume 46, Academic Press, New York and London, 1972. Zbl0246.57017MR54 #1265
  5. [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. Zbl0252.22015MR46 #3689
  6. [Fa] M. FANKHAUSER, Reductive Group Actions on Acyclic Varieties, Thesis, Basel, 1994. 
  7. [Hs] W.-Y. HSIANG, On the Geometric Weight System of Differentiable Compact Transformation Groups on Acyclic Manifolds, Inventiones Math., 12 (1971), 35-47. Zbl0217.49401MR46 #916
  8. [HH70] W.-C. HSIANG and W.-Y. HSIANG, Differentiable actions of compact connected classical groups : II, Annals of Mathematics, 92 (1970), 189-223. Zbl0205.53902MR42 #420
  9. [HH74] W.-C. HSIANG and W.-Y. HSIANG, Differentiable actions of compact connected Lie groups : III, Annals of Mathematics, 99 (1974), 220-256. Zbl0285.57026MR49 #11550
  10. [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. Zbl0476.22010MR83m:57032
  11. [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. Zbl0583.57025MR87m:57041
  12. [Hu] J. E. HUMPHREYS, Linear Algebraic Groups, GTM 21, Springer-Verlag, New York-Heidelberg-Berlin, 1987. 
  13. [Kr84] H. KRAFT, Geometrische Methoden in der Invariantentheorie, Aspekte der Mathematik, band D1, Vieweg, Braunschweig, 1984. Zbl0569.14003MR86j:14006
  14. [Kr89a] H. KRAFT, G-vector bundles and the linearization problem, Contemporary Mathematics, 10 (1989), 111-123. Zbl0703.14009MR90j:14062
  15. [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. Zbl0719.14030MR91g:14044
  16. [KP] H. KRAFT and V.L. POPOV, Semisimple group actions on three dimensional affine space are linear, Comment. Math. Helvetici, 60 (1985), 466-479. Zbl0645.14020MR87a:14039
  17. [KS] H. KRAFT and G. SCHWARZ, Reductive group actions with one-dimensional quotient, Publications Mathématiques IHES, 76 (1992), 1-97. Zbl0783.14026MR94e:14065
  18. [Lu] D. LUNA, Slices étales, Bull. Soc. Math. France, Mémoire 33 (1973), 81-105. Zbl0286.14014MR49 #7269
  19. [Ol] R. OLIVER, Weight systems for SO(3)-actions, Annals of Mathematics, 110 (1979), 227-241. Zbl0465.57017MR80m:57036
  20. [PR] T. PETRIE and J.D. RANDALL, Finite-order algebraic automorphisms of affine varieties, Comment. Math. Helvetici, 61 (1986), 203-221. Zbl0612.14046MR88a:57073
  21. [SK] M. SATO and T. KIMURA, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., 65 (1977), 1-155. Zbl0321.14030MR55 #3341
  22. [Sch] G.W. SCHWARZ, Exotic algebraic group actions, C R. Acad. Sci. Paris, 309 (1989), 89-94. Zbl0688.14040MR91b:14066
  23. [Ve] J.-L. VERDIER, Caractéristique d'Euler-Poincaré, Bull. Soc. Math. France, 176 (1973), 441-445. Zbl0302.57007MR50 #8580
  24. [Vi] È.B. VINBERG, The Weyl Group of a graded Lie Algebra, Izv. Akad. Nauk SSSR, 40 (1976), 463-495. Zbl0371.20041

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