Numerical spectrums control Cohomological spectrums
Junyi Xie
TL;DR
The work addresses how spectral data of a surjective endomorphism on numerical Chow groups compare to spectral data on $l$-adic cohomology for smooth projective varieties. It develops a unified framework of numerical and cohomological spectra, radii, and polygon encodings via cohomological correspondences, enabling precise comparisons. The main result ties the radii $\alpha_j$ and $\beta_i$ through equalities $\log \alpha^-_{2i}(f)=\log \beta^-_i(f)$ and $\log \beta_i(f)=\log \alpha_{2i}(f)$, and shows the numerical and cohomological convex hulls coincide; this extends Deligne's Weil bound and confirms a Tate conjecture aspect for polarized endomorphisms. Consequences include sharp fixed-point counts for int-amplified and $q$-polarized endomorphisms, as well as moving-target variants, and the results generalize to a broad class of cohomological correspondences, bridging geometric dynamics on numerical groups with arithmetic cohomology.
Abstract
Let $X$ be a smooth irreducible projective variety over a field $\mathbf{k}$ of dimension $d.$ Let $τ: \mathbb{Q}_l\to \mathbb{C}$ 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.