Parole chiave

• cohomologically trivial automorphism
• higher elliptic product
• surface automorphism

Abstract For a compact complex manifold $X$, let $\text{Aut}(X)$ denote its group of automorphisms. In the talk I will mainly consider two subgroups of $\text{Aut}(X)$: $\text{Aut}_{\mathbb{Z}}(X)$ the subgroups of cohomologically trivial automorphisms, i.e. of those automorphisms acting trivially on the integral cohomology $H^\ast(X,\mathbb{Z})$; and the larger subgroup $\text{Aut}_{\mathbb{Q}}(X)$ of numerically trivial automorphisms, i.e. of those automorphisms acting trivially on the rational cohomology $H^\ast(X,\mathbb{Q})$. 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 $\text{Aut}_{\mathbb{Z}}(X)$ and $\text{Aut}_{\mathbb{Q}}(X)$ 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 $\geqslant 2$ and $G$ is a finite group acting diagonally. In particular, I will show that in the pseudo-elliptic case ($G$ acts by translations on $E$), $\text{Aut}_{\mathbb{Z}}(X)=E$, or $\vert\text{Aut}_{\mathbb{Z}}(X)/E\vert=2$; while, if $G$ does not act by translations on $E$, then $\text{Aut}_{\mathbb{Z}}(X)$ 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 $\text{Aut}_{\mathbb{Z}}(X)$ and $\text{Aut}_{\mathbb{Q}}(X)$ for some surfaces isogenous to a higher product with $\chi(S) = 1$. In particular, I will describe two surfaces: one having $\vert\text{Aut}_{\mathbb{Q}}(X)\vert=192$, and another one with $\text{Aut}_{\mathbb{Z}}(X)$ of order $2$.