On h-principle and specialness for complex projective manifolds

TL;DR

This paper explores the relationship between the h-principle and the specialness of complex projective manifolds, showing that satisfying the h-principle implies the manifold is special, and discusses related properties like Oka and hyperbolicity.

Contribution

It establishes that complex projective manifolds satisfying the h-principle are necessarily special and raises questions about the converse and extensions to quasi-Kähler cases.

Findings

Manifolds satisfying the h-principle are 'special'

Existence of Stein manifolds with degenerate Kobayashi pseudometric obstructs hyperbolicity

Raises open questions about the reverse implications and broader classes

Abstract

We show that a complex projective manifold X which satisfies the Gromov's h-principle is `special', and raise some questions about the reverse implication, the extension to the quasi-K\" ahler case, and the relationships of these properties to the `Oka' property. The guiding principle is that the existence of many Stein manifolds which have degenerate Kobayashi pseudometric gives strong obstructions to the complex hyperbolicity of X satisfying the h-principle.