Invariant Rings and Quasiaffine Quotients

We study Hilbert's fourteenth problem from a geometric point of view. Nagata's celebrated counterexample demonstrates that for an arbitrary group action on a variety the ring of invariant functions need not be isomorphic to the ring of functions of an affine variety. Nevertheless one can prove that such a ring of invariants is always isomorphic to the ring of functions on a quasi-affine variety. Conversely, for a given quasi-affine variety V there exists always an action of the additive group on some affine variety W such that the ring of functions of V is isomorphic to the ring of invariant functions on W. Thus a k-algebra occurs as invariant ring for some group acting on a k-variety iff it occurs as function ring for some quasi-affine k-variety.Comment: 11 pages, LaTe

A Lie Group without universal covering

We present an example of a disconnected Lie group for which there is no universal covering (as Lie group).Comment: LaTeX, 2 pages. Revised version. A small, but crucial mistake in the formula defining the group structure needed to be correcte

Flat Vector Bundles over Parallelizable Manifolds

We study flat vector bundles over complex parallelizable manifolds

Holomorphic functions on an algebraic group invariant under a Zariski-dense subgroup

Let G be complex linear-algebraic group, H a subgroup, which is dense in G in the Zariski-topology. Assume that G/[G,G] is reductive and furthermore that (1) G is solvable, or (2) the semisimple elements in G'=[G,G] are dense. Then every H-invariant holomorphic function on G is constant. If G=G', furthermore every H-invariant meromorphic or plurisubharmonic function is constant. Finally an example of Margulis is used to show the existence of an algebraic group G with G=G' such that there exists a Zariski-dense discrete subgroup without any semisimple element.Comment: Plain TeX, 9 pages. TeX errors correcte

On tameness and growth conditions

We study discrete subsets of C^d, relating "tameness" with growth conditions.Comment: 6 pages; LaTe

Entire curves, Integral sets and Principal bundles

We compare the behaviour of entire curves and integral sets, in particular in relation to locally trivial fiber bundles, algebraic groups and finite ramified covers over semi-abelian varieties.Comment: LaTeX 19 page

Large Discrete Sets in Stein manifolds

Rosay and Rudin constructed examples of discrete subsets of C^n with remarkable properties. We generalize these constructions from C^n to arbitrary Stein manifolds. We prove: Given a Stein manifold X and a affine variety V of the same dimension there exists a discrete subset D in X such that (1) X-D is measure hyperbolic, (2) f(V) intersects D for every non-degenerate holomorphic map from V to X and (3) every automorphism of X preserving the set D is already the identity map. We also give examples which demonstrate that such discrete subsets can not be found in arbitrary non-Stein manifolds.Comment: LaTex 15 page

Tame discrete subsets in Stein manifolds

For discrete subsets in ${\bf C}^n$ the notion of being "tame" was defined by Rosay and Rudin. We propose a general definition of "tameness" for arbitrary complex manifolds and show that many results classically known for ${\bf C}^n$ may be generalized to semisimple complex Lie groups. For example, every permutation of $SL(2,{\bf Z})$ extends to a biholomorphic self-map of $SL(2,{\bf C}$.Comment: 24 pages, LaTe

Non-linearizable Actions of Commutative Reductive Groups

We generalize a construction of Freudenburg and Moser-Jauslin in order to obtain an example of a non-linearizable action of a commutative reductive algebraic group on the affine space for every field of characteristic zero which admits a quadratic field extension.Comment: 10 pages; revised version, proofs clarifie

Surface Foliations with Compact complex leaves are holomorphic

Let X be a compact complex surface with a real foliation. If all leaves are compact complex curves, the foliation must be holomorphic.Comment: LaTeX, 4 page