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Dominique Cerveau

Publications and source records attributed to Dominique Cerveau.

At least 19 recordsLinked to original sources
arXiv2026

The Geometry of the Exceptional Component of Degree Two Foliations on $\mathbb{P}^3$

We study the exceptional component of the space $\mathbb{F}(2,\mathbb{P}^3)$, of codimension-one foliations of degree two on $\mathbb{P}^3$. We describe the geometry of its boundary and prove that it has four irreducible components, all of dimension $12$. Three of these components contain a dense subset given by the orbit of a logarithmic foliation of type $(1,1,2)$, while the fourth contains a family of pull-back type foliations from $\mathbb{P}^2$ whose orbits have dimension $11$.

math.AG↗
arXiv2024

The Gauss map of a projective foliation

In this paper, we study the Gauss map of a holomorphic codimension one foliation on the projective space $\mathbb{P}^n$, $n\ge 2$, mainly the case $n=3$. Among other things, we will investigate the case where the Gauss map is birational.

math.AG↗
arXiv2023

Singularities of holomorphic codimension one foliations of the complex projective plane

We prove that any holomorphic codimension 1 foliation on the complex projective plane has at most one singular point up to the action of an ad-hoc birational self map of the complex projective plane into itself. Consequently, any algebraic foliation on the affine plane has no singularities up to the action of a suitable birational self map of the complex projective plane into itself.

math.DS↗
arXiv2022

Foliations on $\mathbb{CP}^3$ of degree $2$ that have a line as singular set

In this work we classify foliations on $\mathbb{CP}^3$ of codimension 1 and degree $2$ that have a line as singular set. To achieve this, we do a complete description of the components. We prove that the boundary of the exceptional component has only 3 foliations up to change of coordinates, and this boundary is contained in a logarithmic component. Finally we construct examples of foliations on $\mathbb{CP}^3$ of codimension 1 and degree $s \geq 3$ that have a line as singular set and such that they form a family with a rational first integral of degree $s+1$ or they are logarithmic foliations where some of them have a minimal rational first integral of degree not bounded.

math.AG↗
arXiv2022

Algebraic properties of the group of germs of diffeomorphisms

We establish some algebraic properties of the group $\mathrm{Diff}(\mathbb{C}^n,0)$ of germs of analytic diffeomorphisms of $\mathbb{C}^n$, and its formal completion $\widehat{\mathrm{Diff}}(\mathbb{C}^n,0)$. For instance we describe the commutator of $\mathrm{Diff}(\mathbb{C}^n,0)$, but also prove that any finitely generated subgroup of $\mathrm{Diff}(\mathbb{C}^n,0)$ is residually finite; we thus obtain some constraints of groups that embed into $\mathrm{Diff}(\mathbb{C}^n,0)$. We show that $\widehat{\mathrm{Diff}}(\mathbb{C}^n,0)$ is an Hopfian group, and that $\mathrm{Diff}(\mathbb{C}^n,0)$ and $\widehat{\mathrm{Diff}}(\mathbb{C}^n,0)$ are not co-Hopfian. We end by the description of the automorphisms groups of $\widehat{\mathrm{Diff}}(\mathbb{C},0)$, and $\mathrm{Diff}(\mathbb{C},0)$.

math.GR↗
arXiv2020

The isotropy group of a foliation: the local case

Given a holomorphic singular foliation $\fa$ of $(\C^n,0)$ we define $Iso(\fa)$ as the group of germs of biholomorphisms on $(\C^n,0)$ preserving $\fa$: $Iso(\fa)=\{Φ\in Diff(\C^n,0)\,|\,Φ^*(\fa)=\fa\}$. The normal subgroup of $Iso(\fa)$, of biholomorphisms sending each leaf of $\fa$ into itself, will be denoted as $Fix(\fa)$. The corresponding groups of formal biholomorphisms will be denoted as $\wh{Iso}(\fa)$ and $\wh{Fix}(\fa)$, respectively. The purpose of this paper will be to study the quotients $Iso(\fa)/Fix(\fa)$ and $\wh{Fix}(\fa)/\wh{Fix}(\fa)$, mainly in the case of codimension one foliation.

math.DS↗
arXiv2018

Integrable deformations of foliations: a generalization of Ilyashenko's result

We study analytic deformations of holomorphic differential 1-forms. The initial 1-form is exact homogeneous and the deformation is by polynomial integrable 1-forms. We investigate under which conditions the elements of the deformation are still exact or, more generally, exhibit a first integral. Our results are related to natural extensions of classical results of Ilyashenko on limit cycles of perturbations of hamiltonian systems in two complex variables.

math.AG↗
arXiv2018

Logarithmic Foliations

The purpose of this paper is to study singular holomorphic foliations of arbitrary codimension defined by logarithmic forms on projective spaces.

math.CV↗
arXiv2016

Codimension two holomorphic foliations

This paper is devoted to the study of codimension two holomorphic foliations and distributions. We prove the stability of complete intersection of codimension two distributions and foliations in the local case. Converserly we show the existence of codimension two foliations which are not contained in any codimension one foliation. We study problems related to the singular locus and we classify homogeneous foliations of small degree.

math.DS↗
arXiv2016

Integrable deformations of local analytic fibrations with singularities

We study analytic integrable deformations of the germ of a holomorphic foliation given by $df=0$ at the origin $0 \in \mathbb C^n, n \geq 3$. We consider the case where $f$ is a germ of an irreducible and reduced holomorphic function. Our central hypotheses is that, {\em outside of a dimension $\leq n-3$ analytic subset $Y\subset X$, the analytic hypersurface $X_f : (f=0)$ has only normal crossings singularities}. We then prove that, as germs, such deformations also exhibit a holomorphic first integral, depending analytically on the parameter of the deformation. This applies to the study of integrable germs writing as $ω= df + f η$ where $f$ is quasi-homogeneous. Under the same hypotheses for $X_f : (f=0)$ we prove that $ω$ also admits a holomorphic first integral. Finally, we conclude that an integrable germ $ω= adf + f η$ admits a holomorphic first integral provided that: (i) $X_f: (f=0)$ is irreducible with an isolated singularity at the origin $0 \in \mathbb C^n, n \geq 3$; \, (ii) the algebraic multiplicities of $ω$ and $f$ at the origin satisfy $ν(ω) = ν(df)$. In the case of an isolated singularity for $(f=0)$ the writing $ω= adf + f η$ is always assured so that we conclude the existence of a holomorphic first integral. Some questions related to Relative Cohomology are naturally considered and not all of them answered.

math.CV↗
arXiv2014

Action of the Cremona group on foliations on $\mathbb{P}^2_\mathbb{C}$: some curious facts

The Cremona group of birational transformations of $\mathbb{P}^2_\mathbb{C}$ acts on the space $\mathbb{F}(2)$ of holomorphic foliations on the complex projective plane. Since this action is not compatible with the natural graduation of $\mathbb{F}(2)$ by the degree, its description is complicated. The fixed points of the action are essentially described by Cantat-Favre in \cite{CF}. In that paper we are interested in problems of "aberration of the degree" that is pairs $(ϕ,\mathcal{F})\in\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})\times\mathbb{F}(2)$ for which $\degϕ^*\mathcal{F}<(°\mathcal{F}+1)\degϕ+\degϕ-2$, the generic degree of such pull-back. We introduce the notion of numerical invariance ($\degϕ^*\mathcal{F}=°\mathcal{F}$) and relate it in small degrees to the existence of transversal structure for the considered foliations.

math.DS↗
arXiv2014

Feuilletages holomorphes de codimension 1: une étude locale dans le cas dicritique

Nous décrivons les singularités de feuilletages holomorphes dicritiques de petite multiplicité en dimension $3$. En particulier nous relions l'existence de déformations et de déploiements non triviaux à des problèmes d'intégrabilité liouvillienne. We describe the singularities of dicritical holomorphic foliations of small multiplicity in dimension $3$. In particular we connect the existence of non trivial deformations and deployments to problems of liouvillian integrability.

math.DS↗
arXiv2013

Formes logarithmiques et feuilletages non dicritiques

For a codimension 1 holomorphic foliation $\mathcal F$ on $\mathbb P_{\mathbb C}^{n}$ satisfying reasonable assumptions, there are estimations of the degree of invariant hypersurfaces H in terms of the degree of $\mathcal F$ (Carnicer, Cerveau-Lins Neto). In this paper we study the extremal case $deg H=deg\mathcal F+2$ in the spirit of Brunella's results.

math.DS↗
arXiv2011

Centralisateurs dans le groupe de Jonquières

We give a criterion to determine when the degree growth of a birational map of the complex projective plane which fixes (the action on the basis of the fibration is trivial) a rational fibration is linear up to conjugacy. We also compute the centraliser of such maps. It allows us to describe the centraliser of the birational maps of the complex projective plane which preserve a rational fibration (the action on the basis of the fibration being not necessarily trivial); this question is related to some classical problems of difference equations.

math.AG↗