Equidistribution in Higher Codimension for Holomorphic Endomorphisms of $\mathbb{P}^k$

TL;DR

This paper establishes exponential convergence of certain positive closed currents to the Green current in higher codimension for holomorphic endomorphisms of projective space, under specific smoothness conditions outside an invariant set.

Contribution

It proves exponential equidistribution of positive closed currents to the Green current in higher codimension for holomorphic endomorphisms of projective space, extending previous results.

Findings

Exponential convergence of currents to the Green current.

Existence of a proper invariant analytic subset where smoothness is required.

Results apply to currents smooth outside the invariant set.

Abstract

In this paper, we discuss the equidistribution phenomena for holomorphic endomorphisms over $\mathbb{P}^k$ in the case of bidegree $(p,p)$ with $1<p<k$. We prove that if $f:\mathbb{P}^k\to\mathbb{P}^k$ is a holomorphic endomorphism of degree $d\geq 2$ and $T^p$ denotes the Green $(p,p)$-current associated with $f$, then there exists a proper invariant analytic subset $E$ for $f$ such that $d^{-pn}(f^n)^*(S)\to T^p$ exponentially fast in the current sense for every positive closed $(p,p)$-current $S$ of mass 1 such that $S$ is smooth on $E$.