Applications of a Parametric Oka Principle for Liftings

Lárusson, Finnur

doi:10.1007/978-3-0346-0009-5_12

Cited by 6 publications

(7 citation statements)

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“…CAP was introduced in [14,16]. CIP, introduced in [50], is a restricted version of the Oka property with interpolation (see below). It is easily seen that CIP implies CAP (see [48]), but the converse implication only comes as part of the general theory and no simple proof is known.…”

Section: Oka Manifoldsmentioning

confidence: 99%

“…Gromov's result that ellipticity is equivalent to the Oka property for Stein manifolds has been generalized to a much larger class of manifolds using a geometric structure somewhat more involved than a dominating spray ( [48]; see also [50], §3).…”

Section: Open Problemsmentioning

confidence: 99%

“…Gromov's Oka principle has a natural home in structures provided by abstract homotopy theory. This point of view was developed in the papers [46,47,48,50]. The Oka property of holomorphic maps (and in particular of complex manifolds) turns out to have a natural and rigorous homotopy-theoretic interpretation.…”

Section: Thus a Holomorphic Submersion Of One Of These Two Types Is O...mentioning

confidence: 99%

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Survey of Oka theory

Forstnerič¹,

Lárusson²

2010

Preprint

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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 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 firstnamed author at the conference RAFROT 2010 in Rincón, Puerto Rico, on 22 March 2010, and of a recent survey article by the second-named author [51].

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Section: Oka Manifoldsmentioning

confidence: 99%

Section: Open Problemsmentioning

confidence: 99%

Section: Thus a Holomorphic Submersion Of One Of These Two Types Is O...mentioning

confidence: 99%

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Survey of Oka theory

Forstnerič¹,

Lárusson²

2010

Preprint

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“…The proof Theorem 1.2, proposed in [9], requires the parametric Oka property for sections of certain continuous families of subelliptic submersions. When Finnur Lárusson asked for explanation and at the same time told me of his applications of this result [24] (personal communication, December 2008), I decided to write a more complete exposition. We prove Theorem 1.2 in Sec.…”

mentioning

confidence: 99%

“…5 as a consequence of Theorem 4.2. This result should be compared with Lárusson's [24,Theorem 3] where the map π : E → B is assumed to be an intermediate fibration in the model category that he constructed.…”

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confidence: 99%

Invariance of the parametric Oka property

Forstnerič¹

2009

Preprint

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Assume that E and B are complex manifolds and that 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.Dedicated to Linda P. Rothschild The Oka propertiesThe main result of this paper is that a subelliptic holomorphic submersion π : E → B between (reduced, paracompact) complex spaces satisfies the parametric Oka property. Subellipticity means that E admits a finite dominating family of holomorphic fiber-sprays over a neighborhood of any point in B (Def. 2.3). The conclusion means that for any Stein source space X, any compact Hausdorff space P (the parameter space), and any continuous map f : X × P → B which is X-holomorphic (i.e., such that f p = f (• , p) : X → B is holomorphic for every p ∈ P ), a continuous lifting F : X × P → E of f (satisfying π • F = f ) can be homotopically deformed through liftings of f to an X-holomorphic lifting. (See Theorem 4.2 for a precise statement.

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Invariance of the Parametric Oka Property

Forstnerič

2010

Complex Analysis

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Applications of a Parametric Oka Principle for Liftings

Cited by 6 publications

References 5 publications

Survey of Oka theory

Survey of Oka theory

Invariance of the parametric Oka property

Invariance of the Parametric Oka Property

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