Holomorphic flexibility properties of complex manifolds

Abstract

Abstract. We obtain results on approximation of holomorphic maps by algebraic maps, the jet transversality theorem for holomorphic and algebraic maps between certain classes of manifolds, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds.

Transversality theorems for the weak topology

“…Many transversality theorems have been proved in this case (see Kaliman and Zaidenberg [7] and Forstnerič [3]). It is worth clarifying that in the transversality theorem (theorem 4.2) in [3] we don't need the complex analytic subvarieties to be a-regular.…”

“…Here we first discuss the Thom transversality theorem for the weak topology and then give a similiar kind of result for the weak topology, under very weak hypotheses. Recently several transversality theorems have been proved for complex manifolds and holomorphic maps (see [7] and [3]). In view of these transversality theorems we also prove a result analogous to Trotman's result in the complex case.…”

Approximation of Analytic Sets with Proper Projection by Algebraic Sets

“…Here let us just mention that this theorem is related to the problem of approximation of holomorphic maps between complex (algebraic) spaces, for which the reader is referred to [1,9,13,14,18,19,[25][26][27].…”

Extending holomorphic mappings from subvarieties in Stein manifolds

“…Our results suggest that CAP (and its parametric analogue) is the most natural Oka-type property to be studied further since it is the simplest to verify, yet equivalent to all other Oka properties. Indeed CAP is just the localization of the Oka property with approximation to maps from compact convex sets in Eulidean spaces (see [14] for this point of view). CAP easily follows from ellipticity or subellipticity, and is a natural opposite property to Kobayashi-Eisenman-Brody hyperbolicity [1], [8], [26].…”

“…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].…”