A new geometric construction of compact complex manifolds in any dimension

Abstract

We consider holomorphic linear foliations of dimension m of 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 López 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. (1991): 35J18, 58F18, 32G05 Mathematics Subject Classification

LVMB manifolds and simplicial spheres

“…Finally, a LVM manifold is a manifold constructed as in [16] or [17]. We don't explain here the whole construction of the LVM manifolds.…”

“…The Hopf manifold has many generalizations: firstly, by Calabi and Eckmann [7] who give a structure of complex manifold on any product of spheres (of odd dimension). Then by Santiago Lopez de Medrano, Alberto Verjovsky ( [15] and [16]) and Laurent Meersseman [17]. In these last generalizations, the authors obtain complex structures on products of spheres, and on connected sums of products of spheres, also constructed as a quotient of an open subset in C n but by the action of a non discrete group.…”

“…The construction in [17] has (at least) two interesting features: on the one hand, the LVM manifolds are endowed with an action of the torus (S 1 ) n whose quotient is a simple convex polytope and the combinatorial type of this polytope characterizes the topology of the manifold. On the other hand, for every simple polytope P , it is possible to construct a LVM manifold whose quotient is P .…”

“…In [3], Frédéric Bosio generalizes the construction of [17] emphasing on the combinatorial aspects of LVM manifolds. The aim of this paper is to study the LVMB manifolds (i.e.…”

Moment-angle Complexes, Monomial Ideals and Massey Products

“…Compact, complex, non-Kähler manifolds. Classical constructions of complex manifolds, due to Hopf and Calabi-Eckmann were generalized in recent years by López de Medrano-Verjovsky [39] and Meersseman [38]. These authors define a large class of compact complex manifolds admitting no Kähler structure.…”