A note on automorphisms of the affine Cremona group (Q2511492)

Recall that an ind-group, or infinite dimensional group, is a group that is endowed with a filtration by affine varieties compatible with the group operation. A typical example is provided by the affine Cremona group, or group of polynomial automorphisms of the complex affine space. The main result of this note is as follows. Consider \(G\) an ind-group, \(T \subseteq G\) a closed torus and \(U \subseteq G\) a unipotent ind-subgroup normalized by \(T\). Assume moreover that \(T\) is not in the centralizer of \(u\) for any non-trivial \(u \in U\). Then if \(\theta: G \to G\) is an abstract automorphisms that is the identity in restriction to \(T\), then \(\theta(U)\) is a unipotent ind-subgroup of \(G\). As an application, it is proved that if \(\theta\) is an abstract automorphism of the group Aut\((\mathbb C^3)\) that is the identity in restriction to the tame subgroup, then \(\theta\) also fixes the Nagata automorphism, which is a famous example of non-tame automorphism.