Solvable automorphism groups of varieties

In this talk, I will discuss connected solvable subgroups of the automorphism group $\mathrm{Aut}(X)$ of an arbitrary algebraic variety $X$. When $X$ is projective, these subgroups are well-studied because the automorphism group $\mathrm{Aut}(X)$ is finite-dimensional. However, for general or affine $X$, the problem becomes rather difficult.

The result I will discuss: if $X$ is a quasi-affine variety, then every solvable subgroup of $\mathrm{Aut}(X)$ that is generated by an irreducible algebraic family containing the identity is an algebraic group. As an application, we will see that for a general variety $X$:

- The derived length of a connected solvable subgroup of $\mathrm{Aut}(X)$ is bounded by $\dim(X) + 1$, which implies it is contained in a Borel subgroup.

- Every subgroup of $\mathrm{Aut}(X)$ admits well-defined solvable and unipotent radicals. Furthermore, the unipotent radical is an increasing union of connected unipotent algebraic subgroups.

This is joint work with S. Cantat, H. Kraft and I. van Santen.