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Joerg Winkelmann

Publications and source records attributed to Joerg Winkelmann.

At least 37 records · Page 2Linked to original sources
arXiv2004

Holomorphic discs with dense images

We prove that for any complex manifold X, the set of all holomorphic maps from the unit disc to X whose images are everywhere dense in X forms a dense subset in the space of all holomorphic maps from the disc to X. We show by an example that this need not hold for maps from the disc to complex spaces with singularities.

math.CV↗
arXiv2004

An Example related to Brody's theorem

We discuss an example related to the method of Brody. The example is an open subset of a compact complex torus which is covered by entire curves, but not by Brody curves.

math.CV↗
arXiv2003

Dense Random Finitely Generated Subgroups of Lie Groups

Let G be a connected real Lie group of dimension n. Then there exists a relatively compact open neighbourhood W of e in G such that for n+1 randomly chosen elements g_0,..,g_n the generated subgroup will be dense in G with probability one.

math.GR↗
arXiv2003

Non-degenerate Maps and Sets

We construct certain non-degenerate maps and sets, mainly in the complex-analytic category. For example, we show that for every countable subset S in an irreducible complex space X there exists a holomorphic map from the unit disk to X such that S is contained in the image. Furthermore, given an irreducible complex space X, there is always an infinite subset S such that for every proper analytic subspace Z of X the intersection of S with Z is finite.

math.CV↗
arXiv2003

On Elliptic Curves in SL_2(C)/Γ, Schanuel's conjecture and geodesic lengths

Let H be a discrete cocompact subgroup of SL_2(C). We conjecture that the quotient manifold X=SL_2(C)/H contains infinitely many non-isogeneous elliptic curves and prove that this is indeed the case if Schanuel's conjecture holds. We also prove it in the special case where the intersection of H and SL_2(R) is cocompact in SL_2(R). Furthermore, we deduce some consequences for the geodesic length spectra of real hyperbolic 2- and 3-folds.

math.AG↗
arXiv2002

Bounds for Curves in Abelian Varieties

A uniform bound of intersection multiplicities of curves and divisors on abelian varieties is proved by algebraic geometric methods. It extends and improves a result obtained by A. Buium with a different method based on Kolchin's differential algebra. The problem is modeled after the abc-Conjecture of Masser-Oesterle for abelian varieties over the function field of a curve. As an application a finiteness theorem is proved for maps from a curve into an abelian variety omitting an ample divisor.

math.AG↗
arXiv2002

A note on Jets of Entire Curves in Semi-Abelian Varieties

We prove a product decomposition of the Zariski closure of the jet lifts of a holomorphic map f from C into a semi-abelian variety A, provided that f is of finite order. On the other hand, by giving an example of such a map f into a three dimensional abelian variety we show that this product decomposition does not hold in general; there was a gap in the proofs of some articles which claimed that such a product decomposition is always true.

math.AG↗
arXiv2000

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.

math.AG↗
arXiv1999

The Second Main Theorem for Holomorphic Curves into Semi-Abelian Varieties

Let f:C -> A be an entire holomorphic curve into a semi-Abelian variety A. Then the Zariski closure of f(C) is a translate of a semi-Abelian subvariety of A (logarithmic Bloch-Ochiai's theorem). The purpose of the present paper is to establish a quantitative version of the above result for such f i.e., the second main theorem and the defect relation.

math.CV↗
arXiv1999

Holomorphic Curves and Integral Points off Divisors

We deal with the distributions of holomorphic curves and integral points off divisors. We will simultaneouly prove an optimal dimension estimate from above of a subvariety W off a divisor D which contains a Zariski dense entire holomorphic curve, or a Zariski dense D-integral point set, provided that in the latter case everything is defined over a number field. Then, if the number of components of D is large, the estimate leads to the constancy of such a holomorphic curve or the finiteness of such an integral point set. At the begining, we extend logarithmic Bloch-Ochiai's Theorem to the Kaehler case.

math.CV↗
arXiv1998

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.

math.CV↗