Stein manifolds, a prestack being a contravariant simplicial

Abstract. 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 between Stein manifolds through mapping cylinders in three different model structures and use these factorizations to prove implications between ostensibly different Oka properties of complex manifolds and holomorphic maps. We show that for Stein manifolds, several Oka properties coincide and are characterized by the geometric condition of ellipticity. Going beyond the Stein case to a study of cofibrant models of arbitrary complex manifolds, using the Jouanolou Trick, we obtain a geometric characterization of an Oka property for a large class of manifolds, extending our result for Stein manifolds. Finally, we prove a converse Oka Principle saying that certain notions of cofibrancy for manifolds are equivalent to being Stein. Introduction. In this paper, we apply concepts and tools from abstract homotopy theory to complex analysis and geometry, based on the foundational work in [L2], continuing our development of the idea that the Oka Principle is about fibrancy in suitable model structures. A mapping cylinder in a model category is an object through which a given

Abstract. 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 manifolds, we discuss the recently introduced category of Oka manifolds and Oka maps. We consider geometric sufficient conditions for being Oka, the most important of which is ellipticity, introduced by Gromov. We explain how Oka manifolds and maps naturally fit into an abstract homotopy-theoretic framework. We describe recent applications and some key open problems. This article is a much expanded version of the lecture given by the first-named author

Abstract. We obtain results on approximation of holomorphic maps by algebraic maps, jet transversality theorems for holomorphic and algebraic maps, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds. We also describe the hierarchy between several holomorphic flexibility properties. 1.

Abstract. Assume that E and B are complex manifolds and π: E→B is a holomorphic Serre fibration such that E admits a finite dominating family of holomorphic fiber-sprays over a small neighborhood of any point in B. We show that the parametric Oka property (POP) of B implies POP of E; conversely, POP of E implies POP of B for contractible parameter spaces. This follows from a parametric Oka principle for holomorphic liftings which we establish in the paper. 1. The Oka properties The main result of this paper is the parametric Oka principle for lifting holomorphic sections in subelliptic submersions over a Stein base (Theorem 4.2). This implies that the parametric Oka property of a complex manifold passes up from B to E in a subelliptic Serre fibration π: E → B, and also in a holomorphic fiber bundle whose fiber satisfies the parametric Oka property (Theorem 1.2). The parametric Oka property also passes down from E to B when the parameter space is contractible, or when π is a weak homotopy equivalence.

Abstract. Let Y be a complex manifold with the property that every holomorphic map from a neighborhood of any compact convex set K in a complex Euclidean space C n to Y can be approximated, uniformly on K, by entire maps C n → Y. If X is a reduced Stein space and π: Z → X is a holomorphic fiber bundle with fiber Y then we show that sections X → Z satisfy the Oka principle with approximation and interpolation. The analogous result holds for stratified fiber bundles and also in the parametric case. Dedicated to Professor Joseph J. Kohn on the occasion of his 75th birthday 1.

Abstract. Let Y be a complex manifold with the property that every holomorphic map from a neighborhood of a compact convex set K ⊂ C n to Y can be approximated uniformly on K by entire maps C n → Y. If X is a reduced Stein space and π: Z → X is a holomorphic fiber bundle with fiber Y then we show that sections X → Z enjoy the Oka property with interpolation and approximation. 1.

Abstract. Let Y be a complex manifold with the property that 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 C n → Y. If X is a reduced Stein space and π: Z → X is a holomorphic fiber bundle with fiber Y then we show that sections X → Z satisfy the Oka principle with approximation and interpolation. The analogous result holds for stratified fiber bundles and for submersions with stratified sprays. Dedicated to Joseph J. Kohn on the occasion of his 75th birthday 1.