# Automorphisms of 3-folds of general type acting trivially on cohomology

## TL;DR

This paper investigates automorphisms acting trivially on cohomology for threefolds of general type, establishing bounds on their size and exploring higher-dimensional analogs, revealing cases with arbitrarily large automorphism groups.

## Contribution

It provides new bounds on the size of automorphism groups acting trivially on cohomology for certain threefolds and explores their structure in higher dimensions, including explicit examples.

## Key findings

- Bound of 6 for automorphisms on threefolds of maximal Albanese dimension.
- Bound of 5 for automorphisms on smooth, ample canonical class threefolds.
- Existence of threefolds with arbitrarily large automorphism groups.

## Abstract

Let $X$ be a minimal projective threefold of general type over $\mathbb{C}$ with only Gorenstein quotient singularities, and let $\mathrm{Aut}_{\mathbb{Q}}(X)$ be the subgroup of automorphisms acting trivially on $H^*(X,\mathbb{Q})$. In this paper, we show that if $X$ is of maximal Albanese dimension, then $|\mathrm{Aut}_{\mathbb{Q}}(X)|\leq 6$. Moreover, if $X$ is nonsingular and $K_X$ is ample, then $|\mathrm{Aut}_{\mathbb{Q}}(X)|\leq 5$. Seeking for higher-dimensional examples of varieties with nontrivial $\mathrm{Aut}_{\mathbb{Q}}(X)$, we concern $d$-folds $X$ isogenous to an unmixed product of curves. If $d=3$, we show that $\mathrm{Aut}_{\mathbb{Q}}(X)$ is a $2$-elementray abelian group whose order is at most $4$ under some conditions on their minimal realizations. Moreover, each of the possible groups can be realized. If $d\geq 3$, we give a sufficient condition for $\mathrm{Aut}_{\mathbb{Q}}(X)$ being trivial. Curiously, there exist examples of projective threefolds $X$ with terminal singularities and maximal Albanese dimension whose $\mathrm{Aut}_{\mathbb{Q}}(X)$ can have an arbitrarily large order.