We obtain sufficient and necessary conditions (in terms of positive singular metrics on an associated line bundle) for a positive divisor D on a projective algebraic variety X to be attracting for a holomorphic map f:X → X.

We show that the local automorphism group of a minimal real-analytic CR manifold $M$ is a finite dimensional Lie group if (and only if) $M$ is holomorphically nondegenerate by constructing a jet parametrization.

Let S ⊂ ℂⁿ, n ≥ 3, be a compact connected 2-codimensional submanifold having the following property: there exists a Levi-flat hypersurface whose boundary is S, possibly as a current. Our goal is to get examples of such S containing at least one special 1-hyperbolic point: a sphere with two horns, elementary models and their gluings. Some particular cases of S being a graph are also described.

Every homogeneous circular convex domain $D\subset {\mathbb{C}}^n$ (a bounded symmetric domain) gives rise to two interesting Lie groups: The semi-simple group $G=Aut\left(D\right)$ of all biholomorphic automorphisms of $D$ and its isotropy subgroup $K\subset GL\left(n,\mathbb{C}\right)$ at the origin (a maximal compact subgroup of $G$). The group $G$ acts in a natural way on the compact dual $X$ of $D$ (a certain compactification of ${\mathbb{C}}^n$ that generalizes the Riemann sphere in case $D$ is the unit disk in $\mathbb{C}$). Various authors have studied the orbit structure of the $G$-space $X$, here we are interested...

We construct the CR invariant canonical contact form $can\left(J\right)$ on scalar positive spherical CR manifold $(M,J)$, which is the CR analogue of canonical metric on locally conformally flat manifold constructed by Habermann and Jost. We also construct another canonical contact form on the Kleinian manifold $\Omega \left(\Gamma \right)/\Gamma$, where $\Gamma$ is a convex cocompact subgroup of ${Aut}_{CR}{\text{S}}^{2n+1}=PU(n+1,1)$ and $\Omega \left(\Gamma \right)$ is the discontinuity domain of $\Gamma$. This contact form can be used to prove that $\Omega \left(\Gamma \right)/\Gamma$ is scalar positive (respectively, scalar negative, or scalar vanishing) if and...

For a strongly pseudoconvex domain $D\subset {ℂ}^{n+1}$ defined by a real polynomial of degree $k_0$, we prove that the Lie group $\mathrm{Aut}\left(D\right)$ can be identified with a constructible Nash algebraic smooth variety in the CR structure bundle $Y$ of $\partial D$, and that the sum of its Betti numbers is bounded by a certain constant $C_{n,k_0}$ depending only on $n$ and $k_0$. In case $D$ is simply connected, we further give an explicit but quite rough bound in terms of the dimension and the degree of the defining polynomial. Our approach is to adapt the Cartan-Chern-Moser...

In 1958, H. Grauert proved: If D is a strongly pseudoconvex domain in a complex manifold, then D is holomorphically convex. In contrast, various cases occur if the Levi form of the boundary of D is everywhere zero, i.e. if ∂D is Levi flat. A review is given of the results on the domains with Levi flat boundaries in recent decades. Related results on the domains with divisorial boundaries and generically strongly pseudoconvex domains are also presented. As for the methods, it is explained how Hartogs...

We prove that the Bergman metric on domains satisfying condition S is complete. This implies that any bounded pseudoconvex domain with Lipschitz boundary is complete with respect to the Bergman metric. We also show that bounded hyperconvex domains in the plane and convex domains in ${ℂ}^n$ are Bergman comlete.

We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for CR manifolds, that was first introduced by two of the authors, and the Hörmander’s bracket condition for real vector fields.Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators.Finally we describe a class of compact homogeneous CR manifolds for which the distribution of $(0,1)$ vector fields satisfies...

A normal form for small CR-deformations of the standard CR-structure on the (2n+1)-sphere is presented. The space of normal forms is parameterized by a single function on the sphere. For n>1, the normal form is used to obtain explicit embeddings into ${ℂ}^{n+1}$. For n=1, the cohomological obstruction to embeddability is identified.