| I'll report on recent joint work with Blum, Halpern-Leistner and Xu where we establish that the automorphism group of a K-polystable Fano variety is reductive. In this talk, we will discuss the notion of S-completeness and its connection to reductivity. We will show how the condition of S-completeness for the moduli of log Fanos reduces to the finite generation of a certain ring. |