Relative dynamical degrees of correspondences over a field of arbitrary characteristic

TL;DR

This paper introduces relative dynamical degrees for dominant correspondences over algebraically closed fields of any characteristic, establishing their properties, invariance, and product formulas, even in singular or reducible varieties.

Contribution

It defines and studies relative dynamical degrees for correspondences over arbitrary characteristic fields, extending known results to singular, reducible, and non-algebraically closed cases.

Findings

Defines relative dynamical degrees for correspondences.

Establishes product formulas and birational invariance.

Proves inequalities for semi-conjugacies and generalizations.

Abstract

Let $K$ be an algebraically closed field of arbitrary characteristic, $X$ an irreducible variety and $Y$ an irreducible projective variety over $K$, both are not necessarily smooth. Let $f:X\rightarrow X$ and $g:Y\rightarrow Y$ be dominant correspondences, and $\pi :X\rightarrow Y$ a dominant rational map such that $\pi \circ f=g\circ \pi$. We define relative dynamical degrees $\lambda _p(f|\pi )$ ($p=0,\ldots ,\dim (X)-\dim (Y)$). These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when $Y$ is smooth and $g$ is a multiple of a rational map, and are birational invariants. More generally, a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semi-conjugacy $(\varphi ,\psi )$ from $(X_2,f_2)\rightarrow (Y_2,g_2)$ to $(X_1,f_1)\rightarrow (Y_1,g_1)$ we have $\lambda _p(f_1|\pi _1)\geq \lambda _p(f_2|\pi _2)$ for all $p$. Many of our results are new even when $K=\mathbb{C}$. We make use of de Jong's alterations and Roberts' version of Chow's moving lemma. In the lack of resolution of singularities, the consideration of correspondences is necessary even when $f,g$ are rational maps. The case $K$ is not algebraically closed further requires working with correspondences over reducible varieties.