Related papers: Oka's principle for holomorphic submersions with s…

We give a proof of the following theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., 2 (1989), 851-897). Let Z be a holomorphic fiber bundle over a Stein manifold. If the fiber of Z…

Let X and Y be complex manifolds. One says that maps from X to Y satisfy the Oka principle if the inclusion of the space of holomorphic maps from X to Y into the space of continuous maps is a weak homotopy equivalence. In 1957 H. Grauert…

Let X be a Stein manifold and let Y be a complex manifold which admits a spray in the sense of Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, pp. 851-897 (1989)). We prove that for every closed…

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…

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…

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…

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…

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…

Oka theory has its roots in the classical Oka-Grauert principle whose main result is Grauert's classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds and…

Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion of an elliptic manifold and proved that every continuous map from a Stein manifold to an elliptic manifold is homotopic to a holomorphic map. We show that a much…

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…

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…

This introduction to the homotopy principle in complex analysis and geometry, better known as the Oka theory, is aimed at wide mathematical audience. After a brief historical survey of the h-principle in smooth analysis and geometry, I…

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…

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…

We prove a homotopy theorem for sheaves. Its application shortens and simplifies the proof of many Oka principles such as Gromov's Oka principle for elliptic submersions.

Oka theory has its roots in the classical Oka principle in complex analysis. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. Following a brief review of Stein…

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…

We prove a parametric Oka principle for equivariant sections of a holomorphic fibre bundle $E$ with a structure group bundle $\mathscr G$ on a reduced Stein space $X$, such that the fibre of $E$ is a homogeneous space of the fibre of…

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 $G$…