Abelian groups of K3 type acting on rationally connected varieties

This talk addresses the classification of finite abelian subgroups in the automorphism groups of rationally connected varieties. This is a classical problem, with its origins going back to the late 19th century. An interesting dichotomy arises in dimension two: the finite abelian subgroups of Cremona group of rank 2 can be divided into two types. The first type consists of groups that can act on a Mori fiber space with non-trivial base (that is, on a conic bundle). The second, “exceptional” type, corresponds to elliptic curves with complex multiplication anti-canonically embedded in certain del Pezzo surfaces. We try to extend this observation to higher dimensions. In general, such exceptional abelian groups should originate from highly symmetric Calabi-Yau subvarieties found in birational modifications of the original rationally connected variety. In dimension 3, this role is played by anti- canonically embedded K3 surfaces of higher Picard rank, leading to a complete classification with exactly four exceptional groups. While these groups are realizable, their embedding into the Cremona group of rank 3 remains an open problem. We will also explore the extension problem for finite abelian groups and its connection to the geometry of 4-dimensional Mori fiber spaces.