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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Eigenvalues and dynamical degrees of self-maps on abelian varieties

Author: Fei Hu
Journal: J. Algebraic Geom.
Published electronically: February 7, 2023
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Abstract | References | Additional Information

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 étale cohomology group $H^i_{\acute {\mathrm {e}}\mathrm {t}}(X, \mathbf {Q}_\ell )$ 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.

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Additional Information

Fei Hu
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, Peoples Republic of China; Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway; and Pacific Institute for the Mathematical Sciences, Vancouver, British Columbia V6T1Z4, Canada
MR Author ID: 1086386
ORCID: 0000-0002-3801-1499

Received by editor(s): August 6, 2021
Received by editor(s) in revised form: February 6, 2022
Published electronically: February 7, 2023
Additional Notes: The author was partially supported by a UBC-PIMS Postdoctoral Fellowship and Young Research Talents grant #300814 from the Research Council of Norway
Dedicated: Dedicated to Professor De-Qi Zhang on the occasion of his sixtieth birthday
Article copyright: © Copyright 2023 University Press, Inc.