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

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 $d\ge 2$ 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 $d^{-pn}(f^n{)}^{\ast }(S)\to T^p$ exponentially fast in the current sense for every positive closed $(p,p)$-current $S$ of mass 1 such that $S$ is smooth on $E$.

Corrected an error in the statement of the main theorem and readability has been improved. This paper is based on the work in arXiv:math/0703702 and arXiv:0901.3000 by other authors; for precise estimates, we go over the proofs with modification

References in corpus (2)

• Value distribution for sequences of rational mappings and complex dynamics
• The Jacobian cocycle and equidistribution towards the Green current