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Marcel Nicolau

Publications and source records attributed to Marcel Nicolau.

6 recordsLinked to original sources
arXiv2025

Birationally integrable vector fields on complex projective surfaces

A rational vector field on a complex projective smooth surface $S$ is said to be birationally integrable if it generates, by integration, a one-parameter subgroup of the group $\operatorname{Bir}(S)$ of birational transformations of $S$. We prove that every birationally integrable vector field is regularizable, i.e. birationally conjugated to a holomorphic vector field. Next, we extend this result to any finite-dimensional Lie algebra $\mathfrak g$ of birationally integrable vector fields. This implies that $\mathfrak g$ is naturally included into the Lie algebra of an algebraic subgroup of $\operatorname{Bir}(S)$. Moreover, we obtain a complete birational classification of birationally integrable Lie algebras that are of dimension two or semisimple, exhibiting holomorphic normal forms of them. We also characterize those birationally integrable algebras of rational vector fields that are maximal.

math.AG↗
arXiv2022

On the automorphism group of foliations with geometric transverse structure

Motivated by questions of deformations/moduli in foliation theory, we investigate the structure of some groups of diffeomorphisms preserving a foliation. We give an example of a $C^\infty$ foliation whose diffeomorphism group is not a Lie group in any reasonable sense. On the positive side, we prove that the automorphism group of a transversely holomorphic foliation or a riemannian foliation is a strong ILH Lie goup in the sense of Omori.

math.DG↗
arXiv2015

Foliations and webs inducing Galois coverings

We introduce the notion of Galois holomorphic foliation on the complex projective space as that of foliations whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. First, we establish general criteria assuring that a rational map between projective manifolds of the same dimension defines a Galois covering. Then, these criteria are used to give a geometric characterization of Galois foliations in terms of their inflection divisor and their singularities. We also characterize Galois foliations on $\mathbb P^2$ admitting continuous symmetries, obtaining a complete classification of Galois homogeneous foliations.

math.DS↗
arXiv2010

Deformations of Kahler manifolds with non vanishing holomorphic vector fields

In this article we study compact Kähler manifolds $X$ admitting non-singular holomorphic vector fields with the aim of extending to this setting the classical birational classification of projective varieties with tangent vector fields. We prove that any such a Kähler manifold $X$ admits an arbitrarily small deformation of a particular type which is a suspension over a torus; that is, a quotient of $F\times \mbb C^s$ fibering over a torus $T=\mbb C^s/Λ$. We derive some results dealing with the structure of such manifolds. In particular, we prove an extension of Calabi's theorem describing the structure of compact Kähler manifolds with $c_1(X)=0$ to general Kähler manifolds with non-vanishing vector fields. A complete classification when $X$ is a projective manifold or when $\dim X\leq s+2$ is also given. As an application, it is shown that the study of the dynamics of holomorphic tangent fields on compact Kähler manifolds reduces to the case of rational manifolds.

math.AG↗
arXiv2009

Deformations Feuilletees Des Varietes De Hopf

In this article, we focus on a very special class of foliations with complex leaves whose diffeomorphism type is fixed. They have a unique compact leaf and the noncompact leaves all accumulate onto it. We show that the complex structure along the non-compact leaves is fixed by the complex structure of the compact leaf. Reciprocally, we prove that the complex structure along a non-compact leaf determines the complex structure along the other leaves. We apply these results to the study of foliated deformations of Hopf manifolds, a foliated analogue to the notion of deformation in the large.

math.CV↗