Numerical spectrums control Cohomological spectrums

TL;DR

This paper proves that for polarized endomorphisms on smooth projective varieties, the spectral radii on numerical and cohomological groups coincide, extending Deligne's theorem and confirming Tate's conjecture.

Contribution

It establishes the equality of spectral radii for endomorphisms on numerical and cohomological groups, generalizing Deligne's theorem and confirming Tate's conjecture for polarized endomorphisms.

Findings

Spectral radii on numerical and cohomological groups are equal for all i.

Eigenvalues of f* on cohomology have norm q^{j/2} for q-polarized endomorphisms.

Results have applications in counting fixed points and related dynamical properties.

Abstract

Let $X$ be a smooth irreducible projective variety over a field $\mathbf{k}$ of dimension $d.$ Let $\tau: \mathbb{Q}_l\to \mathbb{C}$ 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 $f^*$ 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 $f^*$ 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.