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On locally trivial \(G_\alpha\)-actions. (English) Zbl 0913.14012

Authors’ summary: If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (theorem 1). We investigate such normal affine algebras in the case of a locally trivial action on a factorial variety. If the variety is a complex affine space and the ring of invariants is isomorphic to a polynomial ring, then the action is conjugate to a translation (theorem 3). Equivalent, if \(\mathbb{C}^n\) is the total space for a principal \(\mathbb{G}_a\)-bundle over some open subset of \(\mathbb{C}^{n-1}\) then the bundle is trivial. On the other hand, there is a locally trivial \(\mathbb{G}_a\)-action on a normal affine variety with non-finitely generated ring of invariants (theorem 2).

MSC:

14L30 Group actions on varieties or schemes (quotients)
32M05 Complex Lie groups, group actions on complex spaces
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