Mathematical Research Letters

Volume 30 (2023)

Number 6

On numerically trivial automorphisms of threefolds of general type

Pages: 1751 – 1785

DOI: https://dx.doi.org/10.4310/MRL.2023.v30.n6.a5

Authors

Zhi Jiang (Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China)

Wenfei Liu (School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, China)

Hang Zhao (School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan, China)

Abstract

$\def\AutQx{\mathrm{Aut}_\mathbb{Q}(X)}$ In this paper, we prove that the group $\AutQx$ of numerically trivial automorphisms are uniformly bounded for smooth projective threefolds X of general type which either satisfy $q(X) \geq 3$ or have a Gorenstein minimal model. If X is furthermore of maximal Albanese dimension, then $\lvert \AutQx \rvert \leq 4$, and the equality can be achieved by an unbounded family of threefolds previously constructed by the third author. Along the way we prove a Noether type inequality for log canonical pairs of general type with the coefficients of the boundary divisor from a given subset $\mathcal{C} \subset (0, 1]$ such that $\mathcal{C} \cup \{1\}$ attains the minimum.

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The first author was supported by National Key Research and Development Program of China (No. 2020YFA0713200), by the NSFC for Innovative Research Groups (No. 12121001), by the Natural Science Foundation of Shanghai (No. 21ZR1404500), and by the NSFC (No. 11871155 and No. 11731004).

The second author was supported by the NSFC (No. 11971399 and No. 11771294) and by the Presidential Research Fund of Xiamen University (No. 20720210006).

Received 28 June 2021

Received revised 7 May 2022

Accepted 4 June 2022

Published 17 July 2024