Eigenvalues and dynamical degrees of self-maps on abelian varieties

TL;DR

This paper proves Truong's conjecture relating cohomological and numerical dynamical degrees for self-maps on abelian varieties, establishing a key link in algebraic dynamics and revealing new eigenvalue properties in prime characteristic.

Contribution

It confirms Truong's conjecture for abelian varieties and introduces a novel parity result on eigenvalues of self-maps in prime characteristic.

Findings

Proves $ ext{chi}_{2k}(f) = ext{lambda}_k(f)$ for abelian varieties.

Establishes a new eigenvalue parity result in prime characteristic.

Enhances understanding of dynamical degrees in algebraic geometry.

Abstract

Let $X$ be a smooth projective variety over an algebraically closed field, and $f\colon X\to X$ a surjective self-morphism of $X$. The $i$-th cohomological dynamical degree $\chi_i(f)$ is defined as the spectral radius of the pullback $f^{*}$ on the \'etale cohomology group $H^i_{\textrm{\'et}}(X, \mathbf{Q}_\ell)$ and the $k$-th numerical dynamical degree $\lambda_k(f)$ as the spectral radius of the pullback $f^{*}$ on the vector space $\mathsf{N}^k(X)_{\mathbf{R}}$ of real algebraic cycles of codimension $k$ on $X$ modulo numerical equivalence. Truong conjectured that $\chi_{2k}(f) = \lambda_k(f)$ for all $0 \le k \le \dim X$ as a generalization of Weil's Riemann hypothesis. We prove this conjecture in the case of abelian varieties. In the course of the proof we also obtain a new parity result on the eigenvalues of self-maps of abelian varieties in prime characteristic, which is of independent interest.