Large discrete sets in Stein manifolds

We construct infinite discrete subsets of Stein manifolds with remarkable properties. These generalize results of Rosay and Rudin on discrete subsets of . Received August 21, 1998; in final form October 11, 1999 / Accepted Published online February 5, 2001     

Discs in Stein manifolds containing given discrete sets

Denote by the open unit disc in . We prove that given a discrete subset S of a connected Stein manifold M there is a proper holomorphic map such that ; if the map f can be chosen to be an embedding. In addition we prove that we can prescribe higher order contacts of with given one dimensional submanifolds in M. Received: 19 June 2000; in final form: 29 November 2000 / Published online: 19 October 2001     

Lagrangian embeddings into subcritical Stein manifolds

In this paper we obtain new topological restrictions on Lagrangian embeddings into subcritical Stein manifolds. We also extend previous results of Gromov, Oh, Polterovich and Viterbo on Lagrangian submanifolds of ℂ ⁿ to the more general case of subcritical Stein manifolds. Research partially supported by the US-Israel Binational Science Foundation grant 1999086.     

Cohomology of locally q-complete sets in Stein manifolds

We prove that if X is a Stein complex manifold of dimension n and Ω???X a locally q-complete open set in X with q?≤?n?2, then the cohomology groups H ^p (Ω , OΩ) vanish if p?≥?q and OΩ is the sheaf of germs of holomorphic functions on Ω.     

Dicritical holomorphic flows on Stein manifolds

We study holomorphic flows on Stein manifolds. We prove that a holomorphic flow with isolated singularities and a dicritical singularity of the form on a Stein manifold with , is globally analytically linearizable; in particular M is biholomorphic to . A complete stability result for periodic orbits is also obtained. Bruno Scárdua: Partially supported by ICTP-Trieste-Italy. Received: 27 September 2006     

Totally skew embeddings of manifolds

We study a version of Whitney’s embedding problem in projective geometry: What is the smallest dimension of an affine space that can contain an n-dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points? This problem is related to the generalized vector field problem, existence of non-singular bilinear maps, and the immersion problem for real projective spaces. We use these connections and other methods to obtain several specific and general bounds for the desired dimension.     

On complex Banach manifolds similar to Stein manifolds

We give an abstract definition, similar to the axioms of a Stein manifold, of a class of complex Banach manifolds in such a way that a manifold belongs to the class if and only if it is biholomorphic to a closed split complex Banach submanifold of a separable Banach space.     

Complete intersections in Stein manifolds

The purpose of this note is to prove some theorems on set theoretic complete intersections in Stein manifolds (or Stein spaces) which are analogous to results in affine algebraic geometry. Due to the Oka principle in Stein theory one gets stronger results. For example any locally complete intersection Y of dimension 3 in a Stein space X with dim X>2 dim Y is a set theoretic₆ complete intersection. A 4-dimensional submanifold of⁶ is a set theoretic complete intersection if sc ₁ ² (Y)=0 for some integer s>0.Der erstgenannte Autor dankt der Alexander-von-Humboldt-Stiftung für ein Stipendium zu einem Gastaufenthalt an der Universität Münster     

On K-spanned embeddings of projective manifolds

We show that h⁰(X, L)≥2k for every k-spanned line bundle on a smooth complete complex surface X.     

Brody quotients of homogeneous Stein manifolds

Homogeneous Stein manifolds admit a complex equivariant quotient with respect to Brody hyperbolicity. As a consequence, simply connected Stein manifolds homogeneous by a solvable Lie group split into a product of a bounded homogenous Stein domain and a complex cell.     

Induced local actions on taut and Stein manifolds

Let act by biholomorphisms on a taut manifold . We show that can be regarded as a -invariant domain in a complex manifold on which the universal complexification of acts. If is also Stein, an analogous result holds for actions of a larger class of real Lie groups containing, e.g., abelian and certain nilpotent ones. In this case the question of Steinness of is discussed.

     

Affine embeddings of flat compact manifolds I

We show that a compact flat manifold has the structure of a real affine variety. Using generators of the affine algebra we constructed embeddings with flat induced metric of all compact flat manifolds with cyclic holonomy group of order 2.     

Holomorphic vector bundles on open subsets of Stein manifolds

LetM be a connected two-dimensional Stein manifold withH ²(M,Z)=0 andS∈M a discrete subset withS≠ Ø. SetX:=M/S. Fix an integerr≥2. Then there exists a rankr vector bundleE onX such that there is no line bundleL onX with a non-zero mapL →E.