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Model structures and the Oka principle
 2004), 203–223. HOLOMORPHIC MAPPINGS 13
"... Stein manifolds, a prestack being a contravariant simplicial ..."
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Stein manifolds, a prestack being a contravariant simplicial
MAPPING CYLINDERS AND THE OKA PRINCIPLE
"... 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 ..."
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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
THE OKA PRINCIPLE FOR SECTIONS OF STRATIFIED FIBER BUNDLES
, 2008
"... 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 fib ..."
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Cited by 5 (3 self)
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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.
INVARIANCE OF THE PARAMETRIC OKA PROPERTY
, 2009
"... 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 fibersprays 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 impl ..."
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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 fibersprays 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.
STEIN NEIGHBORHOODS, HOLOMORPHIC RETRACTIONS, AND EXTENSIONS OF HOLOMORPHIC SECTIONS
, 2007
"... 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 section ..."
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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.
THE OKA PRINCIPLE FOR STRATIFIED FIBER BUNDLES OVER STEIN SPACES
, 2007
"... 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 fi ..."
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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.
HOLOMORPHIC FLEXIBILITY OF COMPLEX MANIFOLDS
, 2004
"... 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 holomorp ..."
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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.
Survey of Oka theory
, 2011
"... 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 Ok ..."
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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 homotopytheoretic framework. We describe recent applications and some key open problems. This article is a much expanded version of the lecture given by the firstnamed author