English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $\mathbb{K}$ be an algebraically closed field, $X$ a smooth projectivevariety 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 cohomologygroups, and $\lambda (f)$ the spectral radius of the pullback$f^*:N^*(X)\rightarrow N^*(X)$. We prove in this paper, by using consequencesof 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 botha birational invariant and \'etale invariant. More general results are provenif either $\mathbb{K}=\overline{\mathbb{F}_p}$ or the Fundamental Conjecture D(numerical equivalence vs homological equivalence) holds. Among other resultsin the paper, we show that if some properties of dynamical degrees, known inthe case $\mathbb{K}=\mathbb{C}$, hold in positive characteristics, then simpleproofs of Weil's Riemann hypothesis follow.

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