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A proper $ {\bf G}\sb a$ action on $ {\bf C}\sp 5$ which is not locally trivial


Authors: James K. Deveney and David R. Finston
Journal: Proc. Amer. Math. Soc. 123 (1995), 651-655
MSC: Primary 14L30; Secondary 14D25
DOI: https://doi.org/10.1090/S0002-9939-1995-1273487-0
MathSciNet review: 1273487
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Abstract: The quotient of a proper holomorphic $ {G_a}$ action on $ {{\mathbf{C}}^n}$ is known to carry the structure of a complex analytic manifold, and in the case of a rational algebraic action, the geometric quotient exists as an algebraic space. An example is given of a proper rational algebraic action for which the quotient is not a variety, and therefore the action is not locally trivial in the Zariski topology.


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  • [1] J. K. Deveney, D. R. Finston, and M. Gehrke, $ {G_a}$ actions on $ {{\mathbf{C}}^n}$, Comm. Algebra 22 (1994), 4977-4988. MR 1285720 (95e:14038)
  • [2] J. K. Deveney and D. R. Finston, Fields of $ {G_1}$ invariants are ruled, Canad. Math. Bull. 37 (1994), 37-41. MR 1261555 (95c:14063)
  • [3] A. van den Essen, Locally finite and locally nilpotent derivations with applications to polynomial flows, morphisms, and $ {G_a}$ actions. II, Report 9206, Catholic Univ., Nijmegen.
  • [4] A. Fauntleroy, Geometric invariant theory for general algebraic groups, Compositio Math. 55 (1985), 63-87. MR 791647 (86m:14008)
  • [5] A. Fauntleroy and A. Magid, Proper $ {G_a}$ actions, Duke Math. J. 43 (1976), 723-729. MR 0417196 (54:5254)
  • [6] H. Matsumura, Commutative ring theory, Cambridge Stud. Adv. Math., vol. 8, Cambridge Univ. Press, Cambridge, 1989. MR 1011461 (90i:13001)
  • [7] J. Ohm, Space curves as ideal theoretic complete intersections, Studies in Algebraic Geometry, MAA Stud. Math., vol. 20, Math. Assoc. Amer., Washington, DC, 1980. MR 589409 (82f:14051)
  • [8] C. S. Seshandri, Quotient spaces modulo reductive algebraic groups, Ann. of Math. 95 (1972), 511-556. MR 0309940 (46:9044)
  • [9] I. R. Shafarevitch, Basic algebraic geometry, Springer-Verlag, Berlin, 1977. MR 0447223 (56:5538)
  • [10] D. Snow, Unipotent actions on affine space, Topological Methods in Algebraic Transformation Groups, Birkhäuser, Boston, 1989. MR 1040863 (91f:14048)
  • [11] L. Tan, An algorithm for explicit generators of the invariants of the basic $ {G_1}$-actions, Comm. Algebra 17 (1989), 565-572. MR 981471 (90j:13018)
  • [12] J. Winkelmann, On free holomorphic C-actions on $ {{\mathbf{C}}^n}$ and homogeneous Stein manifolds, Math. Ann. 286 (1990), 593-612. MR 1032948 (90k:32094)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1273487-0
Article copyright: © Copyright 1995 American Mathematical Society

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