### On the h-principle and specialness for complex projective manifolds

Frédéric Campana, Jörg Winkelmann
2015 Algebraic Geometry
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
more » ... ashi pseudometric gives strong obstructions to the complex hyperbolicity of projective manifolds satisfying the h-principle. The main result is as follows. Main Theorem. Let X be a complex projective manifold satisfying the h-principle. Then (i) the manifold X is special; (ii) every holomorphic map from X to a Brody hyperbolic Kähler manifold is constant. For an arbitrary complex manifold we prove the statements below. Theorem 1.2. Let X be a complex manifold satisfying the h-principle. Then Let us now recall and introduce some notation. Definition 1.3. We say that a complex space X is (i) C-connected if any two points of X can be connected by a chain of entire curves, that is, holomorphic maps from C to X (this property is preserved by passing to unramified coverings and images by holomorphic maps; if X is smooth, this property is also preserved under proper modifications); (ii) Brody-hyperbolic if any holomorphic map h : C → X is constant; (iii) homotopically C-connected if every holomorphic map f → Y from any unramified covering X of X to a Brody-hyperbolic complex space Y induces maps π k (f ) : π k (X ) → π k (Y ) between the respective homotopy groups which are zero for every k > 0. Observe that any holomorphic map f : X → Y between complex spaces is constant if X is C-connected and Y is Brody-hyperbolic. Thus C-connectedness implies weak C-connectedness. Also, any contractible X is homotopically C-connected. There exist projective smooth threefolds which are homotopically C-connected, but not Cconnected. An example can be found in [CW09] . It is easy to verify that every subelliptic manifold X is C-connected. Conversely, all known examples of connected complex manifolds satisfying the h-principle admit a holomorphic homotopy equivalence to a subelliptic complex space. This suggests the following question. Question 1.4. Let X be a complex connected manifold. If X satisfies the h-principle, does this imply that there exists a holomorphic homotopy equivalence between X and a C-connected complex space Z? Since a compact manifold cannot be homotopic to a proper analytic subset for compact manifolds, this question may be reformulated as follows. Question 1.5. Let X be a compact complex connected manifold. If X satisfies the h-principle, does this imply that X is C-connected? Combining Theorem 6.1 with the abelianity conjecture of [Cam04], we obtain the following purely topological conjectural obstruction to the h-principle. Conjecture 1.6. Every projective manifold satisfying the h-principle has an almost abelian fundamental group.