Relations between dynamical degrees, Weil’s Riemann hypothesis and the standard conjectures

Abstract

Let $\mathbb{K}$ be an algebraically closed field, $X$ a smooth projective variety over $\mathbb{K}$ and $f:X\to X$ a dominant regular morphism. Let $N^i(X)$ be the group of algebraic cycles, of codimension $i$, modulo numerical equivalence. Let $\chi (f)$ be the spectral radius of the pullback $f^{\ast }:H^{\ast }(X,{\mathbb{Q}}_l)\to H^{\ast }(X,{\mathbb{Q}}_l)$ on $l$-adic cohomology groups, and $\lambda (f)$ the spectral radius of the pullback $f^{\ast }:N^{\ast }(X)\to N^{\ast }(X)$. We prove in this paper, by using consequences of Deligne’s proof of Weil’s Riemann hypothesis, that $\chi (f)=\lambda (f)$. This answers affirmatively a question posed by Esnault and Srinivas. Consequently, the algebraic entropy $\log \chi (f)$ of an endomorphism is both a birational invariant and étale invariant. More general results are proven if either $\mathbb{K}={\bar{\mathbb{F}}}_p$ or the Standard Conjecture D holds (this applies specially to Abelian varieties). Among other results in the paper, we show that if some properties of dynamical degrees, known in the case $\mathbb{K}=\mathbb{C}$, hold in positive characteristics, then simple proofs of Weil’s Riemann hypothesis follow. More generally, the analogy in positive characteristic of Serre’s famous result on polarized endomorphisms of compact Kähler manifolds also follows.