Finite abelian groups acting on rationally connected threefolds I: Groups of product type

We initiate the study of finite abelian groups that faithfully act on $3$-dimensional rationally connected varieties. We show that these groups can be naturally divided into three types: The groups of product type are finite abelian groups that are products of two groups that belong to the Cremona group of rank~$1$ and $2$, respectively; the groups of K3 type faithfully act on $G\mathbb{Q}$-Fano threefolds $X$ preserving a K3 surface $S\in|-K_X|$ with at worst du Val singularities; the third type consists of groups that act on $G\mathbb{Q}$-Fano threefolds with empty anti-canonical linear system. The classification of groups of product type follows from a result of J. Blanc. For the groups of K3 type, we establish a boundedness result. We also formulate a conjecture regarding the groups of the third type.