Numerical action for endomorphisms

TL;DR

This paper investigates the action of surjective endomorphisms on the numerical divisor group of projective varieties, establishing conditions for hyperbolicity and amplification through new algebraic and spectral concepts.

Contribution

It introduces a novel spectral framework and algebraic notions of positive cycles to analyze endomorphisms' dynamics on divisors.

Findings

f is cohomologically hyperbolic iff it is quasi-amplified

f is amplified iff all subsystems are cohomologically hyperbolic

Introduces spectral and algebraic tools for endomorphism analysis

Abstract

Let $f: X\to X$ be a surjective endomorphism of a projective variety of dimension $d$. The aim of this paper is to study the action of $f$ on the numerical group of divisors. For exmaple, I proved that $f$ is cohomologically hyperbolic if and only if it is quasi-amplified; and it is amplified if and only if every subsystem of $(X,f)$ is cohomologically hyperbolic. For the proofs, I introduced a notion of spectrum in linear algebra for an open and saliant invariant cone. I also introduce a notion of generated (positive) cycles as an algebraic analogy of (positive) closed current.