Related papers: Stationary holomorphic discs and finite jet determ…
We study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. We describe the structure of the set of periodic points in dimension 2. As…
It is proved that dicritical singularities of real analytic Levi-flat sets coincide with the set of Segre degenerate points.
We provide a new way of simultaneously parametrizing arbitrary local CR maps from real-analytic generic manifolds $M\subset {\mathbb C}^N$ into spheres ${\mathbb S}^{2N'-1}\subset {\mathbb C}^{N'}$ of any dimension. The parametrization is…
We classify generic unfoldings of germs of antiholomorphic diffeomorphisms with a parabolic point of codimension 1 (i.e. a double fixed point) under conjugacy. These generic unfolding depend on one real parameter. The classification is done…
We present a new topological method to study the discriminantal loci of an algebraic variety defined in a product of projective spaces. Our approach relies on an efficient use of groupoid to describe the monodromy. As an example, we treat…
We classify the tube domains in C^4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them there are four new…
We use the specialization homomorphism for the birational automorphism group to study finite order birational automorphisms. For a family of varieties over a DVR, we prove that a birational automorphism of order coprime to the residue…
Using the analytic theory of differential equations, we construct examples of formally but not holomorphically equivalent real-analytic Levi nonflat hypersurfaces in $\CC{n}$ together with examples of such hypersurfaces with divergent…
We study the local equivalence problem for real-analytic ($\mathcal{C}^\omega$) hypersurfaces $M^5 \subset \mathbb{C}^3$ which, in coordinates $(z_1, z_2, w) \in \mathbb{C}^3$ with $w = u+i\, v$, are rigid: \[ u \,=\,…
We prove the holomorphic linearizability of germs of biholomorphisms of (C n , 0), fixing the origin, point at which the linear part has nontrivial Jordan blocks under the following assumptions : We first assume the eigenvalues are of…
For a compact convex subset K with non-empty interior in a finite-dimensional vector space, let G be the group of all smooth diffeomorphisms of K which fix the boundary of K pointwise. We show that G is a C^0-regular infinite-dimensional…
In this paper, we prove that if D is a simply-connected domain in C^2 with generic piecewise smooth Levi-flat boundary and non-compact automorphism group, then D is biholomorphic to the bidisc. The proof is based on a careful analysis of…
We are interested in classifying groups of local biholomorphisms (or even formal diffeomorphisms) that can be endowed with a canonical structure of algebraic group up to add extra formal diffeomorphisms. We show that this is the case for…
We construct complete normal forms for $5$-dimensional real hypersurfaces in $\mathbb C^3$ which are $2$-nondegenerate and also of Levi non-uniform rank zero at the origin point ${\bf p} =0$. The latter condition means that the rank of the…
We are interested in characterizing the holonomy maps associated to integral curves of non-degenerate singularities of holomorphic vector fields. Such a description is well-known in dimension 2 where is a key ingredient in the study of…
We give a new proof of Faran's and Lebl's results by means of a new CR-geometric approach and classify all holomorphic mappings from the sphere in $\mathbb C^2$ to Levi-nondegenerate hyperquadrics in $\mathbb C^3$. We use the tools…
We study jets of germs of holomorphic maps between two strongly pseudoconvex domains under the condition that the image of one domain is contained into the other and a given boundary point is (non-tangentially) mapped to a given boundary…
We explicitly determine the structure equations of 5-dimensional Levi 2-nondegenerate CR hypersurfaces, using our recently constructed canonical Cartan connection for this class of CR manifolds. We also give an outline of the basic…
In our earlier articles we studied tube hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In particular, we showed that the vanishing of the CR-curvature of such a hypersurface is equivalent…
We use classical invariant theory to solve the biholomorphic equivalence problem for two families of plane curve singularities previously considered in the literature. Our calculations motivate an intriguing conjecture that proposes a way…