The dual complex of a $G$-variety

Let $G$ be a finite group acting on a smooth projective variety. An important problem is the classification of $G$-varieties up to $G$-birational equivalence. In the talk, a dual complex will be constructed, the highest homology group of which is a $G$-birational invariant. Using this invariant, new examples of varieties with the action of a group $G$ will be constructed, which cannot be $G$-equivariantly rearranged into a projective space. In particular, we demonstrate the non-linearizability of some actions of an abelian group of high rank on smooth hypersurfaces in a projective space of any dimension and degree at least 3.
The talk is based on the paper: Louis Esser, "The dual complex of a $G$-variety".