On abelian extensions of finite abelian subgroups of Cremona groups

Abstract: In this note, we study extension properties of finite abelian subgroups of $\mathrm{Bir}(X)$ where $X$ is a rational (or rationally connected) variety of dimension at most $4$. We are guided by the following question: is it true that if a finite group $G$ faithfully acts on a rationally connected variety of dimension $n$, then $G$ can faithfully act on a terminal Fano variety of dimension $n$? Using algebraic methods, we prove that up to dimension $4$, abelian extensions of finite abelian subgroups of the Cremona group coincide with direct products of such subgroups, with one exception. This result implies a positive answer to the above question up to dimension $4$ in the case of finite abelian groups, modulo a conjectural description of finite abelian subgroups of $\mathrm{Bir}(X)$ where $X$ is a rationally connected threefold.