Let X be a convex domain in ℂⁿ and let E be a convex subset of X. The relative extremal function $u_{E,X}$ for E in X is the supremum of the class of plurisubharmonic functions v ≤ 0 on X with v ≤ -1 on E. We show that if E is either open or compact, then the sublevel sets of $u_{E,X}$ are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.

On montre l’existence d’applications rationnelles $f:{\mathbb{P}}^k\to {\mathbb{P}}^k$ telles que$f$ est algébriquement stable  : pour tout $n\ge 0$, $deg{f}^{\circ n}={(degf)}^n$,il existe un unique courant positif fermé $T$ de bidegré $(1,1)$ vérifiant $f^{*}T=d·T$ et ${\int }_{{\mathbb{P}}^k}T\wedge {\omega }^{k-1}=1$ où $\omega$ est la forme de Fubini-Study sur ${\mathbb{P}}^k$ et$T$ est pluripolaire  : il existe un ensemble pluripolaire $X\subset {\mathbb{P}}^k$ tel que ${\int }_{X}T\wedge {\omega }^{k-1}=1$

Les ensembles polaires dans ${\mathbf{C}}^n$, c’est-à-dire les ensembles où une fonction plurisousharmonique qui n’est pas $-\infty$ identiquement admet cette valeur, apparaissent comme des ensembles exceptionnels dans beaucoup de problèmes en analyse complexe. Par exemple, la croissance d’une fonction plurisousharmonique en une variable $y$ quand une autre variable $x$ est fixée est essentiellement la même pour tout $x$ sauf quand $x$ appartient à un ensemble polaire. Dans l’article un résultat très précis et général de cette...

Let $A$ be a closed polar subset of a domain $D$ in $ℂ$. We give a complete description of the pluripolar hull ${\Gamma }_{D\times ℂ}^{*}$ of the graph $\Gamma$ of a holomorphic function defined on $D\setminus A$. To achieve this, we prove for pluriharmonic measure certain semi-continuity properties and a localization principle.

We prove a disc formula for the weighted Siciak-Zahariuta extremal function $V_{X,q}$ for an upper semicontinuous function q on an open connected subset X in ℂⁿ. This function is also known as the weighted Green function with logarithmic pole at infinity and weighted global extremal function.

We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in ℂⁿ by Lempert and by Lárusson and Sigurdsson.

Given an irreducible algebraic curves $A$ in ${ℂ}^N$, let $m_d$ be the dimension of the complex vector space of all holomorphic polynomials of degree at most $d$ restricted to $A$. Let $K$ be a nonpolar compact subset of $A$, and for each $d=1,2,...,$ choose $m_d$ points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$ in $K$. Finally, let ${\Lambda }_d$ be the $d$-th Lebesgue constant of the array $\left\{A_{dj}\right\}$; i.e., ${\Lambda }_d$ is the operator norm of the Lagrange interpolation operator $L_d$ acting on $C\left(K\right)$, where $L_d\left(f\right)$ is the Lagrange interpolating polynomial for $f$ of degree $d$ at the points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$. Using techniques of pluripotential...

We continue our study of the dynamics of mappings with small topological degree on projective complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic “equilibrium” measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points...

We study the dynamics near infinity of polynomial mappings f in C2. We assume that f has indeterminacy points and is non constant on the line at infinity L∞. If L∞ is f-attracting, we decompose the Green current along itineraries defined by the indeterminacy points and their preimages. The symbolic dynamics that arises is a subshift on an infinite alphabet.

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...

We study the equidistribution of Fekete points in a compact complex manifold. These are extremal point configurations defined through sections of powers of a positive line bundle. Their equidistribution is a known result. The novelty of our approach is that we relate them to the problem of sampling and interpolation on line bundles, which allows us to estimate the equidistribution of the Fekete points quantitatively. In particular we estimate the Kantorovich–Wasserstein distance of the Fekete points...

Let $f:{\mathbb{P}}^k\to {\mathbb{P}}^k$ be a dominating rational mapping of first algebraic degree $\lambda \ge 2$. If $S$ is a positive closed current of bidegree $(1,1)$ on ${\mathbb{P}}^k$ with zero Lelong numbers, we show – under a natural dynamical assumption – that the pullbacks ${\lambda }^{-n}{\left(f^n\right)}^{*}S$ converge to the Green current $T_f$. For some families of mappings, we get finer convergence results which allow us to characterize all $f^{*}$-invariant currents.

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 ${ℂ}^k$.