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 $f:X\rightarrow X$ a dominant regular morphism. Let $N^i(X)$ be the group of algebraic cycles modulo numerical equivalence. Let $\chi (f)$ be the spectral radius of the pullback $f^*:H^*(X,\mathbb{Q}_l)\rightarrow H^*(X,\mathbb{Q}_l)$ on $l$-adic cohomology groups, and $\lambda (f)$ the spectral radius of the pullback $f^*:N^*(X)\rightarrow N^*(X)$. 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 \'etale invariant. More general results are proven if either $\mathbb{K}=\overline{\mathbb{F}_p}$ or the Fundamental Conjecture D (numerical equivalence vs homological equivalence) holds. 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.