23 research outputs found
Classification of absolutely dicritical foliations of cusp type
We give a classification of absolutely dicritical foliations of cusp type,
that is, the germ of singularities of complex foliations in the complex plane
topologically equivalent to the singularity given by the level of the
meromorphic function \frac{y^{2}+x^{3}}{xy}.Comment: 18 page
Normal forms of foliations and curves defined by a function with a generic tangent cone.
International audienceWe first describe the local and global moduli spaces of germs of foliations defined by analytic functions in two variables with p transverse smooth branches, and with integral multiplicities (in the univalued holomorphic case) or complex multiplicities (in the multivalued ''Darboux'' case). We specify normal forms in each class. Then we study on these moduli space the distribution C induced by the following equivalence relation: two points are equivalent if and only if the corresponding foliations have the same analytic invariant curves up to analytical conjugacy. Therefore, the space of leaves of C is the moduli space of curves. We prove that C is rationally integrable. These rational integrals give a complete system of invariants for these generic plane curves, which extend the well-known cross-ratios between branches
Moduli spaces for topologically quasi-homogeneous functions.
International audienceWe consider the topological class of a germ of 2-variables quasi-homogeneous complex analytic function. Each element f in this class induces a germ of foliation (f = constants) and a germ of curve (f = 0). We first describe the moduli space of the foliations in this class and give analytic normal forms. The classification of curves induces a distribution on this moduli space. By studying the infinitesimal generators of this distribution, we can compute the generic dimension of the moduli space for the curves, and we obtain the corresponding generic normal forms
Rigidity for dicritical germ of foliation in the complex plane
We study some kind of rigidity property for dicritical foliation in the
complex plane. In fact, we prove that for a generic dicritical foliation, there
exists deformations of the resolution space which cannot carry any deformation
of the foliation with constant holonomy pseudo-group. This situation never
occurs in the non-dicritical case
Existence of non-algebraic singularities of differential equation
An algebraizable singularity is a germ of a singular holomorphic foliation
which can be defined in some appropriate local chart by a differential equation
with algebraic coefficients. We show that there exists at least countably many
saddle-node singularities of the complex plane that are not algebraizable.Comment: 11 page