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

Abstract

Let $X$ be a minimal projecitve complex $3$-fold of general type with canonical Gorenstein singularities and $\mathrm{Aut}_0(X)$ be the subgroup of automorphisms acting trivially on $H^*(X,\mathbb{Q})$. We prove that if $X$ has maximal Albanese dimension, $|\mathrm{Aut}_0(X)|\leq 6$. Then we concern the class of complex varieties $X=(C_1\times C_2\times\dots\times C_d)/G$ ($d\geq2$) isogenous to a (higher) product of unmixed type with each $q(C_i/G)\geq 1$. The main result of this paper is that if the irregularity $q(X)\geq d+1$, $\mathrm{Aut}_0(X)$ is trivial. Moreover, in the case $d=3$, let $(C_1\times C_2\times C_3)/G$ itself be the minimal realization of $X$ and $K_i$ be the subgroup of $G$ acting identity on $C_i$, we show that if $G$ is abelian, every $K_i$ is cyclic and $q(X) = 3$, then $\mathrm{Aut}_0(X)\cong \mathbb{Z}_2^k$ with $k=0,1,2$. In the end some examples of complex $3$-folds with $\mathrm{Aut}_0(X)\cong \mathbb{Z}_2$ and $\mathbb{Z}_2^2$ are provided