Automorphism groups of complete algebraic varieties

Algebraic groups are interesting in major part by their ability to act on algebraic varieties. However the whole automorphism group of an algebraic variety does not necessarily carry on a structure of an algebraic group. For instance, automorphism groups of affine varieties are usually not algebraic groups, because they are too "big" and, in particular, contain algebraic families of automorphisms of arbitrarily big dimension.
The situation is different for complete varieties: a theorem of Matsumura and Oort (1967) states that the automorphism group $\mathrm{Aut}(X)$ of a complete algebraic variety $X$ is represented by a group scheme of locally finite type so that the identity component $\mathrm{Aut_0}(X)$ is an algebraic group whose Lie algebra consists of the global vector fields on $X$. We present Grothendieck's proof for projective $X$ using Hilbert schemes and consider examples (algebraic curves, Abelian varieties, etc). If $X$ is not uniruled, then $\mathrm{Aut_0}(X)$ is an Abelian variety. On the contrary, $\mathrm{Aut}(X)$ is a linear algebraic group in many important cases: if $X$ is Fano, or the Picard variety $\mathrm{Pic_0}(X)$ is trivial, or $X$ is equipped with an almost transitive action of a connected linear algebraic group. As for the component group $\mathrm{Aut}(X)/\mathrm{Aut_0}(X)$, it is always countable, but not necessarily finite, and even may be not finitely generated (J. Lesieutre, 2016). We shall discuss some of these issues in the talk.
The talk is based on the expository papers of M. Brion "Some structure theorems for algebraic groups" (Proc. Symp. Pure Math. 94, 2017, 53-125; arXiv:1509.03059) and "Notes on automorphism groups of projective varieties" (2018, https://www-fourier.univ-grenoble-alpes.fr/~mbrion/autos_final.pdf)