Abstract
For a compact complex manifold $X$, let $Aut(X)$ denote its group of automorphisms.
In the talk I will mainly consider two subgroups of $Aut(X)$: $Aut_{\mathbb Z}(X)$ the subgroups of cohomologically trivial automorphisms, i.e. of those automorphisms acting trivially on the integral cohomology $H^*(X,\mathbb Z)$; and the larger subgroup $Aut_{\mathbb Q}(X)$ of numerically trivial automorphisms, i.e. of those automorphisms acting trivially on the rational cohomology $H^*(X,\mathbb Q)$. For curves, these 2 subgroups are easily described, but already for surfaces the situation is quite complicated.
After recalling some known results, I will describe $Aut_{\mathbb Z}(X)$ and $Aut_{\mathbb Q}(X)$ for minimal surfaces with Kodaira dimension 1 and $\chi(S) = 0$ (joint work with F. Catanese, C. Gleißner, W. Liu and M. Schütt). These are surfaces isogenous to a higher elliptic product, i.e. free quotients $(C \times E)/G$ where $E$ is an elliptic curve, $C$ is a curve of genus $\geq 2$ and $G$ is a finite group acting diagonally.
In particular, I will show that in the pseudo-elliptic case ($G$ acts by translations on $E$), $Aut_{\mathbb Z}(X)=E$, or $|Aut_{\mathbb Z}(X)/E|=2$; while, if $G$ does not act by translations on $E$, then $Aut_{\mathbb Z}(X)$ is either cyclic of order at most 3, or the Klein group; and exhibit examples of the former cases.
Finally, I will report on a work in progress with F. Catanese and describe $Aut_{\mathbb Z}(X)$ and $Aut_{\mathbb Q}(X)$ for some surfaces isogenous to a higher product with $\chi(S) = 1$. In particular, I will describe two surfaces: one having $|Aut_{\mathbb Q}(X)|=192$, and another one with $Aut_{\mathbb Z}(X)$ of order 2.
