5 problems

Let $\operatorname{Aut}^0(X)=\lim A_n$ be an admissible filtration by irreducible affine subvarieties. Let $G\subseteq\operatorname{Aut}^0(X)$ be nilpotent or solvable and suppose…

Let $X$ be a variety, let $\mathfrak G=\bigcup_i G_i$ be a connected nested ind-subgroup of $\operatorname{Aut}(X)$, and let $T\subseteq\mathfrak G$ be a maximal torus. Let…

Let $\mathfrak G$ be an ind-group, and let $X$ be a variety with automorphism group $\operatorname{Aut}(X)$. The finite-subset criterion. If every finite subset of $\mathfrak G$ (r…

Let $X$ be an affine variety over an algebraically closed field ${\mathbb K}$ of characteristic zero. Let ${\mathcal U}(X)\subset \operatorname{Aut}(X)$ be the subgroup generated b…

Connectedness conjecture for the stabilizers. The groups $G_n$ are connected and hence irreducible for all $n\geq 0$.