When the automorphism group of a projective variety is linear algebraic?

This talk may be regarded as a continuation of my talk on 15-10-2025. As we have seen in that talk, the automorphism group of a projective (even complete) algebraic variety is represented by a group scheme of locally finite type whose identity component is an algebraic group (the Matsumura-Oort theorem, 1967). However the connected automorphism group $\mathrm{Aut}^0(X)$ of a projective variety $X$ may be quite far from linear. For instance, if $X$ is not uniruled, then $\mathrm{Aut}^0(X)$ is an Abelian variety. In fact, in characteristic 0 any connected algebraic group can be realized as $\mathrm{Aut}^0(X)$ for some smooth projective variety $X$ (Brion, 2014).
It is an interesting question under which conditions the group $\mathrm{Aut}^0(X)$ or even the whole automorphism group $\mathrm{Aut}(X)$ is linear algebraic. We shall discuss several necessary or sufficient conditions for that and consider examples. In particular, $\mathrm{Aut}^0(X)$ is a linear algebraic group if the Picard group $\mathrm{Pic(X)}$ is discrete, and $\mathrm{Aut}(X)$ is linear algebraic if $X$ is Fano or equipped with a locally transitive action of a linear algebraic group. (The latter result is due to Fu-Zhang, 2013, in the complex analytic setting and to Brion, 2018, in the algebraic setting.) On the other side, for varieties of general type, $\mathrm{Aut}(X)$ is finite.
In our exposition, we mostly follow the "Notes on automorphism groups of projective varieties" by M. Brion (2018, https://www-fourier.univ-grenoble-alpes.fr/~mbrion/autos_final.pdf).