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A proper \(G_ a\) action on \(\mathbb{C}^ 5\) which is not locally trivial. (English) Zbl 0832.14036

Let \(G_a\) denote the additive group of the field of complex numbers \(\mathbb{C}\). The quotient of a proper holomorphic \(G_a\) action on \(\mathbb{C}^n\) 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 of a proper rational algebraic action on \(\mathbb{C}^5\) is given, where the quotient is not a variety, and therefore the action is not locally trivial in the Zariski topology.
Reviewer: Li Fuan (Beijing)

MSC:

14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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