# Degrees of Iterates of Rational Maps on Normal Projective Varieties

## TL;DR

This paper investigates the behavior of intermediate degrees of iterates of dominant rational maps on normal projective varieties, providing new proofs of submultiplicativity and invariance properties using positivity in numerical cycle spaces.

## Contribution

It offers a new proof of submultiplicativity and birational invariance of degrees, and establishes an algebraic inequality generalizing Siu's inequality for cycles.

## Key findings

- Proves submultiplicativity of degrees of iterates.
- Shows degrees are controlled by the pull-back action norm.
- Provides an algebraic inequality for cycles of arbitrary codimension.

## Abstract

Let X be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. We study the sequence of intermediate degrees of the iterates of a dominant rational selfmap of X, recovering former results by Dinh, Sibony [DS05b], and by Truong [Tru16].Precisely, we give a new proof of the submultiplicativity properties of these degrees and of its birational invariance. Our approach exploits intensively positivity properties in the space of numerical cycles of arbitrary codimension. In particular, we prove an algebraic version of an inequality first obtained by Xiao [Xia15] and Popovici [Pop16], which generalizes Siu's inequality (see [Trap95]) to algebraic cycles of arbitrary codimension. This allows us to show that the degree of a map is controlled up to a uniform constant by the norm of its action by pull-back on the space of numerical classes in X.