Invariance of the parametric Oka property

Abstract

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.

Applications of a parametric Oka principle for liftings

“…A result similar to our next theorem appears in [4]. The analogous result for the basic Oka property is Theorem 3 in [7].…”

“…Theorem 1 (Parametric Oka principle for liftings [4]). Let X and Y be complex manifolds and f : X → Y be a holomorphic map which is either a subelliptic submersion or a holomorphic fibre bundle whose fibre has the parametric Oka property.…”

“…In this note, which is really an addendum to the author's papers [6] and [7], we use a new parametric Oka principle for liftings, very recently proved by Franc Forstnerič [4], to solve several problems left open in those papers. To make this note self-contained would require a large amount of technical background.…”

Affine simplices in Oka manifolds

“…The notion of a holomorphic submersion being subelliptic was defined by Forstnerič [2], generalising the concept of ellipticity due to Gromov [12]. Subellipticity is the weakest currently-known sufficient geometric condition for a holomorphic map to satisfy POPI (see Forstnerič's recently-proved parametric Oka principle for liftings [6]) and for a complex manifold to be Oka. By the influential work of Grauert in [9] and [10], the primary examples of Oka manifolds, to which our results apply, are complex Lie groups and their homogeneous spaces, that is, complex manifolds on which a complex Lie group acts holomorphically and transitively.…”

Affine simplices in Oka manifolds

“…The notion of a holomorphic submersion being subelliptic was defined by Forstnerič [2], generalising the concept of ellipticity due to Gromov [12]. Subellipticity is the weakest currently-known sufficient geometric condition for a holomorphic map to satisfy POPI (see Forstnerič's recently-proved parametric Oka principle for liftings [6]) and for a complex manifold to be Oka.…”