Local regularity of super-potentials and equidistribution of positive closed currents on \({\mathbb {P}}^k\)
type: Item
instance of: scholarly article
title: Local regularity of super-potentials and equidistribution of positive closed currents on \({\mathbb {P}}^k\)
zbMATH Open document ID: 1414.32018
author: Taeyong Ahn
published in: Mathematische Annalen
publication date: 2018-08-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1607.03988
review text: Let $f:\mathbb{P}^k\to\mathbb{P}^k$ be a holomorphic map of algebraic degree $d\geq2$. An important tool in the study of the dynamics of $f$ is its dynamical Green current $T$, defined by $T=\lim_{n\to\infty}d^{-n}(f^n)^\star\omega$, where $\omega$ is the Fubini-Study form on $\mathbb{P}^k$. The author proves the following general equidistribution result towards $T$: There exists an invariant analytic subset $E$ for $f$ such that if $S$ is a positive closed current of bidegree $(p,p)$ and unit mass on $\mathbb{P}^k$ with bounded super-potential near $E$, then $\lim_{n\to\infty}d^{-pn}(f^n)^\star S=T^p$ in the sense of currents. This generalizes an earlier result of the author [Trans. Am. Math. Soc. 368, No. 5, 3359--3388 (2016; Zbl 1418.37075)], where the assumption was that $S$ is smooth near $E$. \par The notion of super-potentials of positive closed currents was introduced and developed by \textit{T.-C. Dinh} and \textit{N. Sibony} [Acta Math. 203, No. 1, 1--82 (2009; Zbl 1227.32024)], who gave many applications for it to complex dynamics in recent years. The notions of local continuity and local boundedness of super-potentials are introduced and studied by the author in the present paper, as main tool in proving the above theorem. In particular, it is shown that if a positive closed current $S$ has bounded super-potential on an open set $W$, then $S$ has zero Lelong number at each point of $W$. The author also shows that $S$ has bounded, resp. continuous, super-potential near an analytic set $E$ if and only if $S$ is a PB, resp. PC, current near $E$. The notions of PB/PC currents are due to Dinh and Sibony [loc. cit.]. \par The author also proves a similar equidistribution result in the case when $f$ is a regular polynomial automorphism of $\mathbb{C}^k$.
reviewed by: Dan Coman
Mathematics Subject Classification ID: 32H50
Mathematics Subject Classification ID: 32U40
Mathematics Subject Classification ID: 32U25
Mathematics Subject Classification ID: 32H02
Mathematics Subject Classification ID: 37F10
zbMATH DE Number: 6923799
zbMATH Keywords: holomorphic endomorphisms of projective space
zbMATH Keywords: positive closed currents
zbMATH Keywords: super-potentials of positive closed currents
MaRDI profile type: Publication
OpenAlex ID: W2962824677
arXiv ID: 1607.03988
cites work: Equidistribution in higher codimension for holomorphic endomorphisms of \(\mathbb {P}^k\)
cites work: Two characterizations of equilibrium measure of an endomorphism of \(P^k(\mathbb{C})\)
cites work: Q4805126
cites work: Analytic multiplicative cocycles over holomorphic dynamical systems
cites work: Green currents for holomorphic automorphisms of compact Kähler manifolds
cites work: Distribution of the values of meromorphic transformations and applications
cites work: Equidistribution towards the Green current for holomorphic maps
cites work: Super-potentials of positive closed currents, intersection theory and dynamics
cites work: Equidistribution speed for endomorphisms of projective spaces
cites work: Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms
cites work: Measures and dynamics on Noetherian spaces
cites work: Multiplicity of holomorphic functions
cites work: Brolin's theorem for curves in two complex dimensions.
cites work: Eigenvaluations
cites work: An invariant measure for rational maps
cites work: Q4309439
cites work: Q4869193
cites work: Q4846449
cites work: Complex differential and integral geometry and curvature integrals associated to singularities of complex analytic varieties
cites work: Equidistribution towards the Green current
cites work: Extending analytic objects
cites work: Q3322903
cites work: Q5566771
cites work: Value distribution for sequences of rational mappings and complex dynamics
cites work: Equidistribution speed towards the Green current for endomorphisms of \(\mathbb P^k\)
DOI: 10.1007/S00208-017-1579-2
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description: scientific article