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 $g^2$, 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 $\text{Sp}(4g,\mathbb{R})$. (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.