Some geometric problems on G-varieties of complexity 1

Let $G$ be a connected, reductive, linear algebraic group that acts on a normal variety $X$, and $B$ be a Borel subgroup group of $G$. The complexity of the $G$-action on $X$ was defined by E. B. Vinberg in 1985 as the codimension of $B$-orbit at a sufficiently general position. $G$-varieties of complexity 0 are the well-known spherical varieties, which have been widely studied by many authors in the past decades. $G$-varieties of complexity 1 also provides rich examples, the basic theory on geometrical structure was founded by D. A. Timashev in 1997.

In the first lecture, I will give an overview on the classification theory, as well as some fundamental theorems on geometrical structure of $G$-varieties of complexity 1 which are mainly established by D. A. Timashev. In the second one, I will introduce some recent works on geometrico-analysis problems on $G$-varieties of complexity 1, such as K-stability and its weighted version. We will start from the former works of Suss, Ilten, Langlois, Terpereau, etc. until our recent ones.