×

zbMATH — the first resource for mathematics

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).
MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] W. Barth andM. Otte, Invariante holomorphe Funktionen auf reduktiven Liegruppen. Math. Ann.201, 97-112 (1973). · Zbl 0253.32018
[2] H. Bass, A non-triangular action ofG a onA 3. J. Pure Appl. Algebra33, 1-5 (1984). · Zbl 0555.14019
[3] E. Cline, B. Parschall andL. Scott, Induced modules and affine quotients. Math. Ann.230, 1-14 (1977). · Zbl 0378.20033
[4] O. Forster, Zur Theorie der Steinschen Algebren und Moduln. Math. Z.97, 376-405 (1967). · Zbl 0148.32203
[5] H. Grauert, On meromorphic equivalence relations. Contributions to Several Complex Variables, Aspects of MathematicsE9, 115-147 (1986). · Zbl 0592.32008
[6] H.Grauert and R.Remmert, Theory of Stein Spaces. Berlin-Heidelberg-New York 1979. · Zbl 0433.32007
[7] F. Grosshans, Observable groups and Hubert’s fourteenth problem. Amer. J. Math.95, 229-253 (1973). · Zbl 0309.14039
[8] D. Hadziev, Some questions in the theory of vector invariants. Math. USSR Sbornik1, 383-396 (1967). · Zbl 0182.05402
[9] G. Hochschild andG. D. Mostow, Representations and representative functions of Lie groups, III. Ann. Math.70, 85-100 (1959). · Zbl 0111.03201
[10] G. Hochschild andG. D. Mostow, On the algebra of representative functions of an analytic group, II. Amer. J. Math.86, 869-887 (1964). · Zbl 0152.01301
[11] J.Humphreys, Linear Algebraic Groups. Berlin-Heidelberg-New York 1975. · Zbl 0325.20039
[12] H.Kraft, Geometrische Methoden in der Invariantentheorie. Aspekte der Mathematik D1, Braunschweig-Wiesbaden 1984. · Zbl 0569.14003
[13] D.Mumford, Geometric Invariant Theory. Ergebnisse der Mathematik34, Berlin-Heidelberg-New York 1982. · Zbl 0504.14008
[14] M. Nagata, On the 14th problem of Hilbert. Amer. J. Math.81, 766-772 (1959). · Zbl 0192.13801
[15] M. Roberts, A note onG-sheaves. Math. Ann.275, 573-582 (1986). · Zbl 0595.32014
[16] C. S. Seshadri, On a theorem of Weitzenböck in invariant theory. J. Math. Kyoto Univ.1, 403-409 (1962). · Zbl 0112.25402
[17] D. Snow, Reductive group actions on Stein spaces. Math. Ann.259, 79-97 (1982). · Zbl 0509.32021
[18] D. Snow, Stein quotients of connected complex Lie groups. Manuscripta Math.50, 185-214 (1985). · Zbl 0582.32020
[19] R. Weitzenböck, Über die Invarianten von linearen Gruppen. Acta Math.58, 230-250 (1932). · Zbl 0004.24301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.