96 research outputs found
Deformations of Oka manifolds
We investigate the behaviour of the Oka property with respect to deformations
of compact complex manifolds. We show that in a family of compact complex
manifolds, the set of Oka fibres corresponds to a G-delta subset of the base.
We give a necessary and sufficient condition for the limit fibre of a sequence
of Oka fibres to be Oka in terms of a new uniform Oka property. We show that if
the fibres are tori, then the projection is an Oka map. Finally, we consider
holomorphic submersions with noncompact fibres
Applications of a parametric Oka principle for liftings
A parametric Oka principle for liftings, recently proved by Forstneric,
provides many examples of holomorphic maps that are fibrations in a model
structure introduced in previous work of ours. We use this to show that the
basic Oka property is equivalent to the parametric Oka property for a large
class of manifolds. We introduce new versions of the basic and parametric Oka
properties and show, for example, that a complex manifold has the basic Oka
property if and only if every holomorphic map to from a contractible
submanifold of extends holomorphically to .Comment: A few minor improvements in version 2. To appear in a volume in
honour of Linda P. Rothschild, Trends in Mathematics series, Birkhause
Approximation of holomorphic mappings on strongly pseudoconvex domains
Let D be a relatively compact strongly pseudoconvex domain in a Stein
manifold, and let Y be a complex manifold. We prove that the set A(D,Y),
consisting of all continuous maps from the closure of D to Y which are
holomorphic in D, is a complex Banach manifold. When D is the unit disc in C
(or any other topologically trivial strongly pseudoconvex domain in a Stein
manifold), A(D,Y) is locally modeled on the Banach space A(D,C^n)=A(D)^n with
n=dim Y. Analogous results hold for maps which are holomorphic in D and of
class C^r up to the boundary for any positive integer r. We also establish the
Oka property for sections of continuous or smooth fiber bundles over the
closure of D which are holomorphic over D and whose fiber enjoys the Convex
approximation property. The main analytic technique used in the paper is a
method of gluing holomorphic sprays over Cartan pairs in Stein manifolds, with
control up to the boundary, which was developed in our paper "Holomorphic
curves in complex manifolds" (Duke Math. J. 139 (2007), no. 2, 203--253)
Stein structures and holomorphic mappings
We prove that every continuous map from a Stein manifold X to a complex
manifold Y can be made holomorphic by a homotopic deformation of both the map
and the Stein structure on X. In the absence of topological obstructions the
holomorphic map may be chosen to have pointwise maximal rank. The analogous
result holds for any compact Hausdorff family of maps, but it fails in general
for a noncompact family. Our main results are actually proved for smooth almost
complex source manifolds (X,J) with the correct handlebody structure. The paper
contains another proof of Eliashberg's (Int J Math 1:29--46, 1990) homotopy
characterization of Stein manifolds and a slightly different explanation of the
construction of exotic Stein surfaces due to Gompf (Ann Math 148 (2):619--693,
1998; J Symplectic Geom 3:565--587, 2005). (See also the related preprint
math/0509419).Comment: The original publication is available at http://www.springerlink.co
An interpolation theorem for proper holomorphic embeddings
Given a Stein manifold X of dimension n>1, a discrete sequence a_j in X, and
a discrete sequence b_j in C^m where m > [3n/2], there exists a proper
holomorphic embedding of X into C^m which sends a_j to b_j for every j=1,2,....
This is the interpolation version of the embedding theorem due to Eliashberg,
Gromov and Schurmann. The dimension m cannot be lowered in general due to an
example of Forster
Every bordered Riemann surface is a complete proper curve in a ball
We prove that every bordered Riemann surface admits a complete proper
holomorphic immersion into a ball of C^2, and a complete proper holomorphic
embedding into a ball of C^3.Comment: Math. Ann., in pres
Obstructions to embeddability into hyperquadrics and explicit examples
We give series of explicit examples of Levi-nondegenerate real-analytic
hypersurfaces in complex spaces that are not transversally holomorphically
embeddable into hyperquadrics of any dimension. For this, we construct
invariants attached to a given hypersurface that serve as obstructions to
embeddability. We further study the embeddability problem for real-analytic
submanifolds of higher codimension and answer a question by Forstneri\v{c}.Comment: Revised version, appendix and references adde
Entire curves avoiding given sets in C^n
Let be a proper closed subset of and
at most countable (). We give conditions
of and , under which there exists a holomorphic immersion (or a proper
holomorphic embedding) with .Comment: 10 page
Families of strictly pseudoconvex domains and peak functions
We prove that given a family of strictly pseudoconvex domains varying
in topology on domains, there exists a continuously varying
family of peak functions for all at every $\zeta\in\partial
G_t.
Flexibility properties in Complex Analysis and Affine Algebraic Geometry
In the last decades affine algebraic varieties and Stein manifolds with big
(infinite-dimensional) automorphism groups have been intensively studied.
Several notions expressing that the automorphisms group is big have been
proposed. All of them imply that the manifold in question is an
Oka-Forstneri\v{c} manifold. This important notion has also recently merged
from the intensive studies around the homotopy principle in Complex Analysis.
This homotopy principle, which goes back to the 1930's, has had an enormous
impact on the development of the area of Several Complex Variables and the
number of its applications is constantly growing. In this overview article we
present 3 classes of properties: 1. density property, 2. flexibility 3.
Oka-Forstneri\v{c}. For each class we give the relevant definitions, its most
significant features and explain the known implications between all these
properties. Many difficult mathematical problems could be solved by applying
the developed theory, we indicate some of the most spectacular ones.Comment: thanks added, minor correction
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