In this paper, we investigate the topology of a class of non-Kahler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics Cn which are invariant with respect to the natural action of the real torus (S1)n onto Cn. The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non-Kahler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kahler situation.

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Real quadrics in Cn, complex manifolds and convex polytopes

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.

A new geometric construction of compact complex manifolds in any dimension

This chapter presents the planar polynomial differential systems. The notion of first integral is introduced; and the definition of integrating factor is discussed. The chapter introduces the notion of an exponential factor due to Christopher. An exponential factor appears when an invariant algebraic curve has in some sense multiplicity larger than 1.

Chapter 5 Integrability of polynomial differential systems

Let X(x,y) and Y(x,y) be real analytic functions without constant and linear terms defined in a neighborhood of the origin. Assume that the analytic differential system \dot{x}=y+ X(x,y), \dot{y}=Y(x,y) has a nilpotent center at the origin. The first integrals, formal or analytic, will be real except if we say explicitly the converse. We prove the following. If X= y f(x,y^2) and Y= g(x,y^2), then the systemhas a local analytic first integral of the form H=y^2+F(x,y),where F starts with terms of order higher than two.If the system has a formal first integral, then it hasa formal first integral of the form H=y^2+F(x,y), where Fstarts with terms of order higher than two. In particular, if thesystem has a local analytic first integral defined at the origin,then it has a local analytic first integral of the formH=y^2+F(x,y), where F starts with terms of order higher than two.As an application we characterize the nilpotent centersfor the differential systems \dot{x}=y+P_3(x,y),\dot{y}=Q_3(x,y), which have a local analytic first integral,where P_3 and Q_3 are homogeneous polynomials of degree three.

Local analytic integrability for nilpotent centers

Branched coverings and algebraic functions

On the topological invariance of the basic cohomology

By using techniques of differential geometry we answer the following open problem proposed by Chavarriga, Giacomini, Gine, and Llibre in 1999. For a given two dimensional flow, what is the maximal order of differentiability of a first integral on a canonical region in function of the order of differentiability of the flow? Moreover, we prove that for every planar polynomial differential system there exist finitely many invariant curves and singular points γ i , i = 1, 2, , l, such that R 2 \ (∪ l i=1 γ i ) has finitely many connected open components, and that on each of these connected sets the system has an analytic first integral. For a homogeneous polynomial differential system in R 3 , there exist finitely many invariant straight lines and invariant conical surfaces such that their complement in R 3 is the union of finitely many open connected components, and that on each of these connected open components the system has an analytic first integral.

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On the differentiability of first integrals of two dimensional flows

Holomorphic flows and complex structures on products of odd dimensional spheres

A class of C ∞ -stable foliations

Let F be a transversely holomorphic foliation on a compact manifold. We show the existence of a versal space for those deformations of F which keep fixed its differentiable type if F is Hermitian or if F has complex codimension one and admits a transverse projectable connection. We also prove the existence of a versal space of deformations for the complex structures on a Lie group invariant by a cocompact subgroup.

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Marcel Nicolau

Papers

On the topological invariance of the basic cohomology

On the differentiability of first integrals of two dimensional flows

Holomorphic flows and complex structures on products of odd dimensional spheres

A class of C ∞ -stable foliations

Déformations des feuilletages transversalement holomorphes à type différentiable fixé