Let $\varphi$ be a psh function on a bounded pseudoconvex open set $\Omega \subset {ℂ}^n$, and let $ℐ\left(m\varphi \right)$ be the associated multiplier ideal sheaves, $m\in {\mathbb{N}}^{☆}$. Motivated by global geometric issues, we establish an effective version of the coherence property of $ℐ\left(m\varphi \right)$ as $m\to +\infty$. Namely, given any $B⋐\Omega$, we estimate the asymptotic growth rate in $m$ of the number of generators of $ℐ{\left(m\varphi \right)}_{|B}$ over ${𝒪}_{\Omega }$, as well as the growth of the coefficients of sections in $\Gamma (B,ℐ(m\varphi \left)\right)$ with respect to finitely many generators globally defined on $\Omega$. Our approach relies on proving asymptotic integral...

Let $f$ be a non-invertible holomorphic endomorphism of a projective space and $f^n$ its iterate of order $n$. We prove that the pull-back by $f^n$ of a generic (in the Zariski sense) hypersurface, properly normalized, converges to the Green current associated to $f$ when $n$ tends to infinity. We also give an analogous result for the pull-back of positive closed $(1,1)$-currents and a similar result for regular polynomial automorphisms of ${ℂ}^k$.

Let $K\subset Q$ be compact, convex sets in ${ℝ}^n$ with $\stackrel{\circ }{K}\ne \varnothing$ and let $P\left(D\right)$ be a linear, constant coefficient PDO. It is characterized in various ways when each zero solution of $P\left(D\right)$ in the space $ℰ\left(K\right)$ of all $C^{\infty }$-functions on $K$ extends to a zero solution in $ℰ\left(Q\right)$ resp. in $ℰ\left({ℝ}^n\right)$. The most relevant characterizations are in terms of Phragmén-Lindelöf conditions on the zero variety of $P$ in ${ℂ}^n$ and in terms of fundamental solutions for $P\left(D\right)$ with lacunas.

We study different notions of extremal plurisubharmonic functions.

A plurisubharmonic singularity is extreme if it cannot be represented as the sum of non-homothetic singularities. A complete characterization of such singularities is given for the case of homogeneous singularities (in particular, those determined by generic holomorphic mappings) in terms of decomposability of certain convex sets in ℝⁿ. Another class of extreme singularities is presented by means of a notion of relative type.