Equidistribution of varieties for endomorphisms of projective spaces

TL;DR

This paper surveys the asymptotic behavior of preimages of algebraic subvarieties under iterates of non-invertible holomorphic endomorphisms of complex projective spaces, focusing on their equidistribution properties.

Contribution

It provides a comprehensive overview of the known results and techniques regarding the equidistribution of preimages of varieties under such endomorphisms.

Findings

Preimages of generic subvarieties tend to become equidistributed as n increases.

The survey summarizes key theorems and conjectures in the field.

It highlights the role of dynamical degrees and potential theory in understanding equidistribution.

Abstract

Let f be a non-invertible holomorphic endomorphism of the complex projective space P^k and f^n its iterate of order n. Let V be an algebraic subvariety of P^k which is generic in the Zariski sense. We give here a survey on the asymptotic equidistribution of the sequence $f^{-n}(V)$ when n goes to infinity.