# zbMATH — the first resource for mathematics

Quotient dimensions and the fixed point problem for reductive groups. (English) Zbl 0834.14027
Summary: Let $$X$$ be a smooth, affine complex variety, which, considered as a complex manifold, has the singular $${\mathbb{Z}}$$-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 $$\pi :X\to X/ / G$$ contains a dense orbit.
If $$G$$ is connected and reductive, then the action has fixed points if $$\dim X/ / G\leq 3$$.

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations
##### Keywords:
action of complex algebraic group; quotient; fixed points
Full Text:
##### References:
 [1] H. BASS, A non-triangular action of ga on A3, Jour. Pure Appl. Alg., 33 (1984), 1-5. · Zbl 0555.14019 [2] 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. · Zbl 0034.30701 [3] N. BOURBAKI, Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968. [4] G.E. BREDON, Introduction to compact transformation groups, pure and applied mathematics, Volume 46, Academic Press, New York and London, 1972. · Zbl 0246.57017 [5] 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. · Zbl 0252.22015 [6] M. FANKHAUSER, Reductive group actions on acyclic varieties, Thesis, Basel, 1994. [7] W.-Y. HSIANG, On the geometric weight system of differentiable compact transformation groups on acyclic manifolds, Inventiones Math., 12 (1971), 35-47. · Zbl 0217.49401 [8] W.-C. HSIANG and W.-Y. HSIANG, Differentiable actions of compact connected classical groups : II, Annals of Mathematics, 92 (1970), 189-223. · Zbl 0205.53902 [9] W.-C. HSIANG and W.-Y. HSIANG, Differentiable actions of compact connected Lie groups : III, Annals of Mathematics, 99 (1974), 220-256. · Zbl 0285.57026 [10] W.-Y. HSIANG and E. STRAUME, Actions of compact connected Lie groups with few orbit types, J. Reine Angew. Math., 334 (1982), 1-26. · Zbl 0476.22010 [11] 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. · Zbl 0583.57025 [12] J. E. HUMPHREYS, Linear algebraic groups, GTM 21, Springer-Verlag, New York-Heidelberg-Berlin, 1987. [13] H. KRAFT, Geometrische methoden in der invariantentheorie, Aspekte der Mathematik, band D1, Vieweg, Braunschweig, 1984. · Zbl 0569.14003 [14] H. KRAFT, G-vector bundles and the linearization problem, Contemporary Mathematics, 10 (1989), 111-123. · Zbl 0703.14009 [15] 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. · Zbl 0719.14030 [16] H. KRAFT and V.L. POPOV, Semisimple group actions on three dimensional affine space are linear, Comment. Math. Helvetici, 60 (1985), 466-479. · Zbl 0645.14020 [17] H. KRAFT and G. SCHWARZ, Reductive group actions with one-dimensional quotient, Publications Mathématiques IHES, 76 (1992), 1-97. · Zbl 0783.14026 [18] D. LUNA, Slices étales, Bull. Soc. Math. France, Mémoire 33 (1973), 81-105. · Zbl 0286.14014 [19] R. OLIVER, Weight systems for SO(3)-actions, Annals of Mathematics, 110 (1979), 227-241. · Zbl 0465.57017 [20] T. PETRIE and J.D. RANDALL, Finite-order algebraic automorphisms of affine varieties, Comment. Math. Helvetici, 61 (1986), 203-221. · Zbl 0612.14046 [21] M. SATO and T. KIMURA, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., 65 (1977), 1-155. · Zbl 0321.14030 [22] G.W. SCHWARZ, Exotic algebraic group actions, C R. Acad. Sci. Paris, 309 (1989), 89-94. · Zbl 0688.14040 [23] J.-L. VERDIER, Caractéristique d’Euler-Poincaré, Bull. Soc. Math. France, 176 (1973), 441-445. · Zbl 0302.57007 [24] È.B. VINBERG, The Weyl group of a graded Lie algebra, Izv. Akad. Nauk SSSR, 40 (1976), 463-495. · Zbl 0371.20041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.