Related papers: J-holomorphic discs and real analytic hypersurface…

Let (M,p) be a smooth non-Leviflat CR hypersurface germ in complex Euclidean space of dimension 2 where p is of infinite type. The purpose of this article is to investigate the holomorphic vector fields tangent to (M,p) vanishing at p.

The Borel map $j^{\infty}$ takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. It is well known that the restriction of $j^{\infty}$ to the germs of quasianalytic ultradifferentiable classes which are…

We explicitly describe germs of strongly pseudoconvex non-spherical real-analytic hypersurfaces $M$ at the origin in $\CC^{n+1}$ for which the group of local CR-automorphisms preserving the origin has dimension $d_0(M)$ equal to either…

Given a holomorphic Lie algebroid $(V, \phi)$ on a compact connected Riemann surface $X$, we give a necessary and sufficient condition for a holomorphic vector bundle $E$ on $X$ to admit a holomorphic Lie algebroid connection. If $(V,…

Let $M\subset \mathbb C^n$ be a real analytic hypersurface, $M'\subset \mathbb C^N$ $(N\geq n)$ be a strongly pseudoconvex real algebraic hypersurface of the special form and $F$ be a meromorphic mapping in a neighborhood of a point $p\in…

We continue our study, initiated in an earlier article, of a class of rigid hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1, having zero CR-curvature. We drop the restrictive assumptions of…

Let M be a real analytic manifold modeled on a locally convex space and K be a non-empty compact subset of M. We show that if an open neighborhood of K in M admits a complexification which is a regular topological space, then the germ of…

We classify all (locally) homogeneous Levi non-degenerate real hypersurfaces in $\mathbb{C}^3$ with symmetry algebra of dimension $\geq 6$.

In this paper, we study the real hypersurfaces $M$ in $\mathbb C^2$ at points $p\in M$ of infinite type. The degeneracy of $M$ at $p$ is assumed to be the least possible, namely such that the Levi form vanishes to first order in the CR…

Let $D$ be a disk in $\mathbb{R}^n$ and $f\in C^{r+2}(D, \mathbb{R}^k)$. We deal with the problem of the algebraic approximation of the set $j^{r}f^{-1}(W)$ consisting of the set of points in the disk $D$ where the $r$-th jet extension of…

The main result establishes an estimate for the growth of a real meromorphic function $f$ on the unit disc $\Delta$ such that: (i) at least one of $f$ and $1/f$ has finitely many poles and non-real zeros in $\Delta$; (ii)~$f^{(k)}$ has…

In this paper, we provide {\em effective} results on the non-embeddability of real-analytic hypersurfaces into a hyperquadric. We show that, for any $N >n \geq 1$, the defining functions $\varphi(z,\bar z,u)$ of all real-analytic…

Let $\mathcal{F}$ be a codimension one holomorphic foliation in $\mathbb{P}^n$, $n\geq 2$, leaving invariant a real-analytic Levi-flat hypersurface $M$ with regular part $M^{*}$. Then every leaf of $\mathcal{F}$ outside $\overline{M^{*}}$…

We consider a continuous function $f$ on a domain in $\mathbf C^n$ satisfying the inequality that $|\bar \partial f|\leq |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable…

Let $H \subset {\mathbb P}^n$ be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian $G(n+1,n)$. Assuming $H$ has a global defining function, we prove $H$ is Levi-flat, the closure of its…

For a smooth, non-degenerate locally integrable structure of hypersurface type on a manifold $M$, we provide necessary and sufficient conditions for it to be equivalent, near a point, to a real-analytic locally integrable structure (the…

It is shown that that the rank of the second fundamental form (resp. the Levi form) of a $\mathcal C^2$-smooth convex hypersurface $M$ in $\Bbb R^{n+1}$ (resp. $\Bbb C^{n+1}$) does not exceed an integer constant $k<n$ near a point $p\in M,$…

In this paper we consider unramified coverings of the moduli space $\mathcal{M}_g$ of smooth projective complex curves of genus $g$. Under some hypothesis on the branch locus of the finite extended map to the Deligne-Mumford…

We study an analytically irreducible algebroid germ (X, 0) of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the divisorial valuations associated to the irreducible…

We introduce new biholomorphic invariants for real-analytic hypersurfaces in 2-dimensional complex space and show how they can be used to show that a hypersurface possesses few automorphisms. We give conditions, in terms of the new…