Green currents of holomorphic correspondences on compact K\"ahler manifolds

TL;DR

This paper constructs Green currents for holomorphic correspondences on compact Kähler manifolds, analyzes their regularity, and proves exponential equidistribution under certain conditions.

Contribution

It introduces a method to construct Green currents associated with dominant eigenspaces and establishes their regularity and equidistribution properties.

Findings

Green currents are associated with dominant eigenspaces of the action on cohomology.

Super-potentials of these Green currents are log-Hölder continuous.

Under specific conditions, positive closed currents equidistribute exponentially towards the Green current.

Abstract

Consider a holomorphic correspondence $f$ on a compact K\"ahler manifold $X$ of dimension $k$. Let $1\le q\le k$ be any integer such that the dynamical degrees of $f$ satisfy $d_{q-1}<d_q$. We construct the Green currents $T_c$ of $f$ associated with the classes $c$ belonging to the dominant eigenspace for the action of $f^*$ on $H^{q,q}(X,\mathbb{R})$. We also show that the super-potential of $T_c$ is $\log$-H\"older-continuous. When $f$ has simple action on cohomology and its graph satisfies an assumption on the local multiplicity, we prove the exponential equidistribution of all positive closed currents towards the main Green current, i.e., the only one associated to the unique maximal degree $d_q$.