A new geometric construction of compact complex manifolds in any dimension

Abstract

We consider holomorphic linear foliations of dimension m of \(\mathbb C^n\) (with \(n>2m\)) fulfilling a so-called weak hyperbolicity condition and equip the projectivization of the leaf space (for the foliation restricted to an adequate open dense subset) with a structure of compact, complex manifold of dimension \(n-m-1\). We show that, except for the limit-case \(n=2m+1\) where we obtain any complex torus of any dimension, this construction gives non-symplectic manifolds, including the previous examples of Hopf, Calabi-Eckmann, Haefliger (linear case), Loeb-Nicolau (linear case) and Lopez de Medrano-Verjovsky. We study some properties of these manifolds, that is to say meromorphic functions, holomorphic vector fields, forms and submanifolds. For each manifold, we construct an analytic space of deformations of dimension \(m(n-m-1)\) and show that, under some additional conditions, it is universal. Lastly, we give explicit examples of new compact, complex manifolds, in particular of connected sums of products of spheres and show the existence of a momentum-like map which classifies these manifolds, up to diffeomorphism.