The local Phragmén-Lindelöf condition for analytic varieties in complex n-space was introduced by Hörmander and plays an important role in various areas of analysis. Recently, new necessary geometric properties for a variety satisfying this condition were derived by the present authors. These results are now applied to investigate the homogeneous polynomials P with real coefficients that are stable in the following sense: Whenever f is a holomorphic function that is defined in some neighborhood...

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...