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Invariants of holomorphic affine flows. (English) Zbl 0612.14009
Weitzenböck’s theorem says that the algebra of polynomials invariant under a linear algebraic representation of the additive group \({\mathbb{C}}\) is finitely generated [R. Weitzenböck, Acta Math. 58, 231-293 (1932; Zbl 0004.24301)]. In the present paper, the author generalizes Weitzenböck’s theorem to holomorphic actions of \({\mathbb{C}}\) as a group of affine transformations on a complex vector space V. He shows that the algebra of invariant polynomials is finitely generated, so that there exists a normal affine variety W and an invariant algebraic morphism \(\pi: V\to W\) which is universal with respect to invariant algebraic morphisms to affine varieties. He also proves that there is a Stein space Z and an invariant holomorphic map \(\tau: V\to Z\) which is universal with respect to invariant holomorphic maps to Stein spaces. The two quotients are connected by a holomorphic map \(\sigma: Z\to W\) such that \(\pi =\sigma \circ \tau\). Moreover, if the action has a fixed point (i.e. is isomorphic to a linear action) then \(Z\cong W\) under \(\sigma\) ; if the action has no fixed points, then Z is isomorphic to a hyperplane in V and V is equivariantly isomorphic to \({\mathbb{C}}\times Z\). Along the way, the author proves that there exists a universal quotient for the action of a maximal unipotent subgroup of a reductive group action on a Stein space, generalizing the corresponding result for algebraic actions on affine varieties (for a discussion of this result and further references see H. Kraft, ”Geometrische Methoden in der Invariantentheorie” (1984; Zbl 0569.14003).
14L24 Geometric invariant theory
32M05 Complex Lie groups, group actions on complex spaces
32E10 Stein spaces
14L30 Group actions on varieties or schemes (quotients)
37C10 Dynamics induced by flows and semiflows
Full Text: DOI
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