AI 中文总结
本文研究了阿贝尔簇导出范畴自等价的动力学，证明了 Gromov--Yomdin 等式、熵上界、平移数刻画及约化平移数有限性条件，并否证了范畴熵函数的代数性问题。
AI 中文摘要
我们建立了关于阿贝尔簇的范畴动力学的若干一般性结果。(1) 我们证明了任意阿贝尔簇的导出范畴的每个自等价都满足指数和多项式 Gromov--Yomdin 等式。指数等式推广了 Kikuta 对椭圆曲线以及 Yoshioka 对阿贝尔曲面和单阿贝尔簇的早期结果。(2) 我们证明了每个自等价的范畴多项式熵以 $g^2$ 为上界，其中 $g$ 是阿贝尔簇的维数。这给出了 Lin--Oguiso--Zhang 的多项式对数体积增长的一个范畴推广，并直接恢复了他们的上界。(3) 我们将阿贝尔簇的每个自等价的平移数等同于 $\text{Sp}(4g,\mathbb{R})$ 的万有覆盖上的规范化 Barge--Ghys 辛平移数的拉回。(4) 我们证明了约化平移数集合是有限的当且仅当 nef 锥是有理多面体。此外，当该集合无限时，存在一个具有超越平移数的自等价。特别地，这给出了对 Dimitrov--Haiden--Katzarkov--Kontsevich 关于范畴熵函数代数性问题的否定回答。
英文摘要
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.
Comments37 pages; comments are welcome
