Related papers: An Oka Principle for a Parametric Infinite Transit…

Given a Stein manifold with the density property, we show that under a suitable topological condition it is possible to prescribe derivatives at a finite number of points to automorphisms depending holomorphically on a Stein parameter. This…

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…

We prove closing lemmas for automorphisms of a Stein manifold with the density property and for endomorphisms of an Oka-Stein manifold. In the former case we need to impose a new tameness condition. It follows that hyperbolic periodic…

In this paper we establish a basic version of the Oka principle for multivalued sections of ramified holomorphic maps h from a complex manifold Z onto a Stein manifold X. If the ramification locus of h projects into a closed complex…

A complex manifold Y satisfies the Convex Approximation Property (CAP) if every holomorphic map from a neighborhood of a compact convex set K in a complex Euclidean space C^n to Y can be approximated, uniformly on K, by entire maps from C^n…

We take the first step in the development of an equivariant version of modern, Gromov-style Oka theory. We define equivariant versions of the standard Oka property, ellipticity, and homotopy Runge property of complex manifolds, show that…

We give the following positive answer to Gromov's question (in "Oka's principle for holomorphic sections of elliptic bundles", J. Amer. Math. Soc. 2, 851-897 (1989), 3.4.(D), page 881). THEOREM: If every holomorphic map from a compact…

Let $X$ be a smooth open manifold of even dimension, $T$ be a topological space, and $\mathscr{J}=\{J_t\}_{t\in T}$ be a continuous family of smooth integrable Stein structures on $X$. Under suitable additional assumptions on $T$ and…

Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold. This gives a positive answer to the well-known long-standing problem in Oka theory whether…

We apply concepts and tools from abstract homotopy theory to complex analysis and geometry, continuing our development of the idea that the Oka Principle is about fibrancy in suitable model structures. We explicitly factor a holomorphic map…

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…

Let $X$ be a connected Oka manifold, and let $S$ be a Stein manifold with $\mathrm{dim} S \geq \mathrm{dim} X$. We show that every continuous map $S\to X$ is homotopic to a surjective strongly dominating holomorphic map $S\to X$. We also…

Let $X$ be a Stein manifold of dimension $n\geq 2$ satisfying the volume density property with respect to an exact holomorphic volume form. For example, $X$ could be $\mathbb{C}^n$, any connected linear algebraic group that is not…

Given a holomorphic submersion of reduced complex spaces, we prove that the basic Oka property of the submersion implies the parametric Oka property. This generalizes the corresponding result for complex manifolds (F. Forstneric, Oka…

Let $n>1$ be an integer. We prove that holomorphic maps from Stein manifolds $X$ of dimension $<n$ to the complement $\mathbb{C}^n\setminus L$ of a compact convex set $L\subset\mathbb{C}^n$ satisfy the basic Oka property with approximation…

Suppose that Y is a complex manifold with the property that any holomorphic map from a compact convex set in a complex Euclidean space C^n (for any n) to Y is a uniform limit of entire maps from C^n to Y. We prove that a holomorphic map…

Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\colon X\to Q_X$ and $p_Y\colon Y\to Q_Y$ be the quotient mappings. Assume that we have a biholomorphism $Q:= Q_X\to Q_Y$ and an open…

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…

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…

We introduce the notion of a stratified Oka manifold and prove that such a manifold $X$ is strongly dominable in the sense that for every $x\in X$, there is a holomorphic map $f:\C^n\to X$, $n=\dim X$, such that $f(0)=x$ and $f$ is a local…