Is a $K3$ surface a spherical variety?

Question

Do there exist any $K3$ surfaces that admit the structure of a spherical variety?

That is, does there exists a $K3$ surface $X$, a reductive algebraic group $G$, and a Borel subgroup $B$ of $G$, such that $X$ admits the structure of a $G$ variety with an open dense $B$ orbit?

Answer 1

Let $X/\bar{k}$ denote a K3 surface. Since $H^0(X, \mathcal{T}_X) = 0$, the automorphism group $\text{Aut}(X)$ is reduced and discrete. Therefore, if an algebraic group $G$ acts on $X$, the induced homomorphism $G \rightarrow \text{Aut}(X)$ must have an open (hence finite index) kernel. In particular, the orbits of $G$ are all finite. Therefore, there is no way for an algebraic group to act on $X$ with a dense orbit.

Reference: Huybrechts, Lectures on K3 surfaces, Chapter 15.