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...

We establish a lower estimate for the Kobayashi-Royden infinitesimal pseudometric on an almost complex manifold $(M,J)$ admitting a bounded strictly plurisubharmonic function. We apply this result to study the boundary behaviour of the metric on a strictly pseudoconvex domain in $M$ and to give a sufficient condition for the complete hyperbolicity of a domain in $(M,J)$.

We show that every small resolution of a 3-dimensional terminal hypersurface singularity can occur on a non-embeddable $1$-convex manifold.We give an explicit example of a non-embeddable manifold containing an irreducible exceptional rational curve with normal bundle of type $(1,-3)$. To this end we study small resolutions of $c{D}_4$-singularities.

It is shown that on strongly pseudoconvex domains the Bergman projection maps a space $L{v}_k$ of functions growing near the boundary like some power of the Bergman distance from a fixed point into a space of functions which can be estimated by the consecutive power of the Bergman distance. This property has a local character. Let Ω be a bounded, pseudoconvex set with C³ boundary. We show that if the Bergman projection is continuous on a space $E\supset L^{\infty }\left(\Omega \right)$ defined by weighted-sup seminorms and equipped with the topology...

We find regular Stein neighborhoods of a union of totally real planes M = (A+iI)ℝ² and N = ℝ² in ℂ², provided that the entries of a real 2 × 2 matrix A are sufficiently small. A key step in our proof is a local construction of a suitable function ρ near the origin. The sublevel sets of ρ are strongly Levi pseudoconvex and admit strong deformation retraction to M ∪ N.

In contrast with the integrable case there exist infinitely many non-integrable homogeneous almost complex manifolds which are strongly pseudoconvex at each boundary point. All such manifolds are equivalent to the Siegel half space endowed with some linear almost complex structure.We prove that there is no relatively compact strongly pseudoconvex representation of these manifolds. Finally we study the upper semi-continuity of the automorphism group of some hyperbolic strongly pseudoconvex almost...

Let $X$ be a complex manifold with strongly pseudoconvex boundary $M$. If $\psi$ is a defining function for $M$, then $-log\psi$ is plurisubharmonic on a neighborhood of $M$ in $X$, and the (real) 2-form $\sigma =i\partial \overline{\partial }(-log\psi )$ is a symplectic structure on the complement of $M$ in a neighborhood of $M$ in $X$; it blows up along $M$. The Poisson structure obtained by inverting $\sigma$ extends smoothly across $M$ and determines a contact structure on $M$ which is the same as the one induced by the complex structure. When $M$ is compact, the Poisson structure near...

We prove the existence of stationary discs in the ball for small almost complex deformations of the standard structure. We define a local analogue of the Riemann map and establish its main properties. These constructions are applied to study the local geometry of almost complex manifolds and their morphisms.

Let $X_1$ and $X_2$ be two compact strongly pseudoconvex CR manifolds of dimension $2n-1\ge 5$ which bound complex varieties $V_1$ and $V_2$ with only isolated normal singularities in ${ℂ}^{N1}$ and ${ℂ}^{N2}$ respectively. Let $S_1$ and $S_2$ be the singular sets of $V_1$ and $V_2$ respectively and $S_2$ is nonempty. If $2n-N_2-1\ge 1$ and the cardinality of $S_1$ is less than 2 times the cardinality of $S_2$, then we prove that any non-constant CR morphism from $X_1$ to $X_2$ is necessarily a CR biholomorphism. On the other hand, let $X$ be a compact strongly pseudoconvex CR manifold of...

Toeplitz operators on strongly pseudoconvex domains in Cn, constructed from the Bergman projection and with symbol equal to a positive power of the distance to the boundary, are considered. The mapping properties of these operators on Lp, as the power of the distance varies, are established.