On the h-principle and specialness for complex projective manifolds

We show that a complex projective manifold X which satisfies Gromov's h-principle is special in the sense of the first author's paper "Orbifolds, Special Varieties, and Classification Theory" (Annals of the Institut Fourier, 2004), and raise some questions about the reverse implication, the extension to the quasi-Kähler case, and the relationships of these properties to the Oka property.The guiding principle is that the existence of many Stein manifolds with degenerate Kobayashi pseudometric gives strong obstructions to the complex hyperbolicity of projective manifolds satisfying the h-principle.