On the numerically and cohomologically trivial automorphisms of elliptic surfaces II: $\chi(S)>0$

Abstract

In this second part we study first the group $Aut_{\mathbb Q}(S)$ of numerically trivial automorphisms of an algebraic properly elliptic surface $S$, that is, of a minimal algebraic surface with Kodaira dimension $\kappa(S)=1$, in the case $\chi(S) \geq 1$. Our first surprising result is that, against what has been believed for over 40 years, there exist nontrivial such groups for $p_g(S) >0$. Indeed, we show even that $Aut_{\mathbb Q}(S)$ is always a 2-generated finite abelian group, but there is no absolute upper bound for its cardinality. At any rate, we give explicit and essentially optimal upper bounds for $|Aut_{\mathbb Q}(S)|$ in terms of the numerical invariants of $S$, as $\chi(S)$, or the irregularity $q(S)$, or the bigenus $P_2(S)$. Moreover, we reach an almost complete description of the possible groups $Aut_{\mathbb Q}(S)$ and we give effective criteria for such surfaces to have trivial $Aut_{\mathbb Q}(S)$. Our second surprising results concern the quite elusive group $Aut_{\mathbb Z}(S)$ of cohomologically trivial automorphisms; we are able to give the explicit upper bounds for $|Aut_{\mathbb Z}(S)|$ in special cases: 9 when $p_g(S) =0$, and we achieve the sharp upper bound 3 when $S$ (i.e., the pluricanonical elliptic fibration) is isotrivial. Also in the non isotrivial case we produce subtle examples where $Aut_{\mathbb Z}(S)$ is a group of order 2 or 3.