Dao Phuong Bac

Let $G$ be a linear algebraic group acting on an affine variety $V$, both
defined over a field $k$, and let $v \in V(k)$ be a rational point. We
denote by $\overline{k}$ the algebraic closure of $k$. Motivated by moduli
problems in Algebraic Geometry, the Zariski closedness of the geometric
orbit $G(\overline{k}).v$ has played a central role in Geometric Invariant
Theory (GIT) since the pioneering work of D. Mumford in the 1960s. Since
2005, G. Roehrle and collaborators have introduced a GIT framework for
analyzing completely reducible subgroups (and their relative version)
proposed by J.-P. Serre, through the concept of cocharacter closedness of
rational orbits $G(k).v$. Here we say that the rational orbit $G(k).v$ of a
rational point $v$ is cocharacter closed if this orbit contains the limit
point (if exists) along any cocharacter of $G$. Now, assuming further that
$k$ is a valued field, we may endow $G(k)$ and $V(k)$ with the $v$-adic
topology induced from the base field $k$. The aim of this talk is to
discuss the relationship between the cocharacter closedness and Hausdorff
closedness of rational orbits, as well as their respective closures in the
cases that $G$ is any torus or the conjugate action of general linear
groups. This is a joint work with Vu Tuan Hien, Vo Duy Hoang, and Nguyen
Minh Duc.