Related papers: Oka manifolds

We generalize our elliptic characterization of Oka manifolds to Oka maps. The generalized characterization can be considered as an affirmative answer to the relative version of Gromov's conjecture. As an application, we unify previously…

In this paper we obtain a Carleman approximation theorem for maps from Stein manifolds to Oka manifolds. More precisely, we show that under suitable complex analytic conditions on a totally real set $ M $ of a Stein manifold $X$, every…

We show that the group of all holomorphic automorphisms of complex affine space $\mathbb C^n$, $n>1$, and several of its subgroups satisfy the parametric Oka property with approximation and with interpolation on discrete sets.

Gromov, in his seminal 1989 paper on the Oka principle, proved that every continuous map from a Stein manifold into an elliptic manifold is homotopic to a holomorphic map. Previously we have shown that, given a continuous map $X \to…

This is a survey on the homotopy principle in complex analysis on Stein manifolds, also called the Oka principle in this context. We concentrate on the following topics: the Oka-Grauert principle (classification of holomorphic vector…

The name of Oka principle, or Oka-Grauert principle, is traditionally used to refer to the holomorphic incarnation of the homotopy principle: on a Stein space, every problem that can be solved in the continuous category, can be solved in…

The basic result of Oka theory, due to Gromov, states that every continuous map $f$ from a Stein manifold $S$ to an elliptic manifold $X$ can be deformed to a holomorphic map. It is natural to ask whether this can be done for all $f$ at…

We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group $G$…

Let $(E,h)$ be a semipositive hermitian holomorphic line bundle on a compact complex manifold $X$ with $\dim X>1$. Assume that for each point $x\in X$ there exists a divisor $D\in |E|$ in the complete linear system determined by $E$ whose…

A complex manifold $Y$ is said to have the interpolation property if a holomorphic map to $Y$ from a subvariety $S$ of a reduced Stein space $X$ has a holomorphic extension to $X$ if it has a continuous extension. Taking $S$ to be a…

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…

The Oka principle is a heuristic in complex geometry which states that, for a wide class of complex-analytic problems concerning Stein spaces, any obstruction to finding a holomorphic solution is purely topological. A classical theorem of…

A complex manifold $X$ satisfies the Oka-Grauert property if the inclusion $\Cal O(S,X) \hookrightarrow \Cal C(S,X)$ is a weak equivalence for every Stein manifold $S$, where the spaces of holomorphic and continuous maps from $S$ to $X$ are…

Let \(\overline \Omega\) be a compact strongly pseudoconvex domain with smooth boundary in a Stein manifold, and let \(h:Z\to \overline \Omega\) be a fibre bundle of H\"older-Zygmund class \(\Lambda^r\), \(r>0\), which is holomorphic over…

Over the past decade, the class of Oka manifolds has emerged from Gromov's seminal work on the Oka principle. Roughly speaking, Oka manifolds are complex manifolds that are the target of "many" holomorphic maps from affine spaces. They are…

In this paper we prove results on the existence and homotopy classification of holomorphic submersions from Stein manifolds to other complex manifolds. We say that a complex manifold Y satisfies Property S_n for some integer n bigger or…

In this note, we show that the homotopy type of a complex manifold X satisfying the Oka property is captured by holomorphic maps from the affine spaces C^n, n\geq 0, into X. Among such X are all complex Lie groups and their homogeneous…

The paper is related to the author's talk at the Hayama Symposium in Complex Analysis in December 2000. In section 1 we survey results on the Oka principle for sections of holomorphic submersions over Stein manifolds. In section 2 we apply…

In this paper we begin a systematic study of the class of complex manifolds which are universal targets of holomorphic maps from open Riemann surfaces. We call them Oka-1 manifolds, by analogy with Oka manifolds that are universal targets…

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…