Numerical spectrums control Cohomological spectrums

Abstract

Let $X$ be a smooth irreducible projective variety over a field $\mathbf{k}$ of dimension $d.$ Let $\ddot{I}„:{\mathbb{Q}}_l\to \mathbb{C}$ be any field embedding. Let $f:X\to X$ be a surjective endomorphism. We show that for every $i=0,\dots ,2d$, the spectral radius of $f^{\ast }$ on the numerical group $N^i(X)\otimes \mathbb{R}$ and on the $l$-adic cohomology group $H^{2i}(X_{\overline{\mathbf{k}}},{\mathbb{Q}}_l)\otimes \mathbb{C}$ are the same. As a consequence, if $f$ is $q$-polarized for some $q>1$, we show that the norm of every eigenvalue of $f^{\ast }$ on the $j$-th cohomology group is $q^{j/2}$ for all $j=0,\dots ,2d.$ This generalizes Deligne's theorem for Weil's Riemann Hypothesis to arbitary polarized endomorphisms and proves a conjecture of Tate. We also get some applications for the counting of fixed points and its ``moving target" variant. Indeed we studied the more general actions of certain cohomological coorespondences and we get the above results as consequences in the endomorphism setting.

18 pages