Invariants of holomorphic affine flows
Weitzenböck's theorem says that the algebra of polynomials invariant under a linear algebraic representation of the additive group \({\mathbb{C}}\) is finitely generated [\textit{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 \textit{H. Kraft}, ''Geometrische Methoden in der Invariantentheorie'' (1984; Zbl 0569.14003).
scientific article; zbMATH DE number 3989514
scientific article