Oka's principle for holomorphic fiber bundles with sprays

Abstract

The subject of this paper is the homotopy principle, also called the h-principle or the Oka-Grauert principle, concerning sections of certain holomorphic fiber bundles on Stein manifolds. We give a proof of a theorem of Gromov (1989) from sec. 2.9 in [Gro]; see theorems 1. 3 and 1.4 below. This result, which extends the work of H. Grauert from 1957 ([Gr3], [Gr4], [Car]), has been used in the proofs of the embedding theorem for Stein manifolds into Euclidean spaces of minimal dimension [EGr], [Sch].1.1 Definition. Let h: Z → X be a holomorphic mapping of complex manifolds. A section of h is any map f : X → Z such that h• f is the identity on X. We say that sections of h satisfy the h-principle (or the Oka-Grauert principle) if each continuous section f 0 : X → Z can be deformed to a holomorphic section f 1 : X → Z through a continuous one parameter family (a homotopy) of continuous sections f t : X → Z (0 ≤ t ≤ 1), and any two holomorphic sections f 0 , f 1 : X → Z which are homotopic through continuous sections are also homotopic through holomorphic sections. If this holds for a trivial bundle Z = X × F → X, we say that maps X → F satisfy the h-principle.1.2 Definition. (Gromov [Gro]) A (dominating) spray on a complex manifold F is a holomorphic vector bundle p: E → F , together with a holomorphic map s: E → F , such that s is the identity on the zero section F ⊂ E, and for each x ∈ F the derivative Ds(x) maps E x (which is a linear subspace of T x E) surjectively onto T x F .The following result can be found in sec. 2.9 of [Gro].1.3 Theorem. If F is a complex manifold which admits a spray, then the sections of any locally trivial holomorphic fiber bundle with fiber F over any Stein manifold satisfy the h-principle. In particular, mappings from Stein manifolds into F satisfy the h-principle.Stronger results are given in theorem 1.4 and corollary 1.5 below. In the sequel [FP] to this paper we give a proof of Gromov's Main Theorem ([Gro], sect. 4.5) to the effect that the h-principle holds for sections of holomorphic submersions h: Z → X, where X is Stein and each point x ∈ X has a neighborhood U ⊂ X such that Z|U = h −1 (U ) admits a fiber-spray (see def. 3.1 below).For non-specialists we recall that a complex manifold is called Stein (after Karl Stein, 1951 [Ste]) if it has 'plenty' of global holomorphic functions. For the precise definition

The homotopy principle in complex analysis: a survey

“…Our main references are the papers by Grauert [Gra1,Gra2], Cartan [Ca], Gromov [Gro3], and [FP1,FP2,FP3,F5].…”

Extending holomorphic mappings from subvarieties in Stein manifolds

“…Each of the above conditions implies CAP (see the h-Runge theorems proved in [9], [15], [21]). The converse implication CAP ⇒ subellipticity is not known in general, and there are cases when CAP is known to hold but the existence of a dominating spray (or a dominating family of sprays) is unclear; see corollary 6.2 below and the examples in [13] and [14].…”

Holomorphic submersions from Stein manifolds

“…The following result in the smooth category was proved by A. Phillips [P] and M. Gromov [Gr1,Gr3] [F4]. For the Oka-Grauert-Gromov theory we refer to [G3,Gr4,HL2,FP1,FP2,FP3,F3,W]. An Theorem 2.2 is proved in Section 4.…”

“…The quotient projection 7r: [A, Fo] [G1, G2]; see also [Gr4] and [FP1].) Applying Theorem 4.1 in [F3] (or Theorem 5.1 in [FP1] [FP1]).…”

“…Applying Theorem 4.1 in [F3] (or Theorem 5.1 in [FP1] [FP1]). An alternative method, proposed by Gromov [Gr4] and developed in [FP2] …”