On numerically trivial automorphisms of threefolds of general type

TL;DR

This paper establishes bounds on the group of numerically trivial automorphisms of certain threefolds of general type, revealing conditions under which these automorphisms are uniformly limited in size.

Contribution

It proves uniform bounds for automorphisms of threefolds of general type under specific conditions, and introduces a Noether type inequality for log canonical pairs.

Findings

Automorphism groups are uniformly bounded for certain threefolds.

Maximum order of automorphisms is at most 4 for threefolds of maximal Albanese dimension.

Constructed examples achieve the maximum automorphism order of 4.

Abstract

In this paper, we prove that the group $\mathrm{Aut}_\mathbb{Q}(X)$ 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 $|\mathrm{Aut}_\mathbb{Q}(X)|\leq 4$, and 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.