Automorphisms of threefolds of general type acting trivially in cohomology

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 $$|\mathrm {Aut}_{\mathbb {Q}}(X)|\le 6$$ . Moreover, if X is nonsingular and $$K_X$$ is ample, then $$|\mathrm {Aut}_{\mathbb {Q}}(X)|\le 5$$ . 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 $$d\ge 3$$ , 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.