Categorical Dynamics of Abelian Varieties
Abstract
We establish several general results on the categorical dynamics of abelian varieties. (1) We prove the exponential and polynomial Gromov--Yomdin equalities for every autoequivalence of the derived category of an arbitrary abelian variety. The exponential equality extends earlier results of Kikuta for elliptic curves and of Yoshioka for abelian surfaces and simple abelian varieties. (2) We prove the categorical polynomial entropy of every autoequivalence is bounded above by g2, where g is the dimension of the abelian variety. This gives a categorical generalization of the polynomial log-volume growth of Lin--Oguiso--Zhang and recovers their upper bound as a direct consequence. (3) We identify the shifting number of every autoequivalence of an abelian variety with the pullback of the normalized Barge--Ghys symplectic translation number on the universal cover of Sp(4g,ℝ). (4) We prove that the set of reduced shifting numbers is finite if and only if the nef cone is rational polyhedral. Moreover, when this set is infinite, there exists an autoequivalence with transcendental shifting number. In particular, this gives a negative answer to a question of Dimitrov--Haiden--Katzarkov--Kontsevich concerning the algebraicity of categorical entropy functions.
Related
- Orbits in stability manifolds and categorical entropy: applications to varieties with finite Albanese morphisms
- Dynamical degrees of automorphisms on abelian varieties
- Categorical Torelli theorem for smooth Fano varieties
- A holomorphic (2, 2)-Theorem for Abelian varieties of CM-type
- Arithmetic jet spaces and the unramified Zilber--Pink conjecture for subvarieties of abelian varieties
BibTeX
@misc{fan2026categorical,
title = {{Categorical Dynamics of Abelian Varieties}},
author = {Yu-Wei Fan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02807},
eprint = {2610.02807},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02807},
}
RIS
TY - UNPB
TI - Categorical Dynamics of Abelian Varieties
AU - Yu-Wei Fan
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02807
UR - https://arxiv.org/abs/2610.02807
AB - We establish several general results on the categorical dynamics of abelian varieties. (1) We prove the exponential and polynomial Gromov--Yomdin equalities for every autoequivalence of the derived category of an arbitrary abelian variety. The exponential equality extends earlier results of Kikuta for elliptic curves and of Yoshioka for abelian surfaces and simple abelian varieties. (2) We prove the categorical polynomial entropy of every autoequivalence is bounded above by g^2, where g is the dimension of the abelian variety. This gives a categorical generalization of the polynomial log-volume growth of Lin--Oguiso--Zhang and recovers their upper bound as a direct consequence. (3) We identify the shifting number of every autoequivalence of an abelian variety with the pullback of the normalized Barge--Ghys symplectic translation number on the universal cover of Sp(4g,ℝ). (4) We prove that the set of reduced shifting numbers is finite if and only if the nef cone is rational polyhedral. Moreover, when this set is infinite, there exists an autoequivalence with transcendental shifting number. In particular, this gives a negative answer to a question of Dimitrov--Haiden--Katzarkov--Kontsevich concerning the algebraicity of categorical entropy functions.
ER -