On $G$-Rigid Surfaces

Rigid algebraic varieties form an important class of complex varieties that exhibit interesting geometric phenomena. In this paper we propose a natural extension of rigidity to complex projective varieties with a finite group action ($G$-varieties) and focus on the first nontrivial case, namely, on $G$-rigid surfaces that can be represented as desingularizations of Galois coverings of the projective plane with Galois group $G$. We obtain local and global $G$‑rigidity criteria for these $G$-surfaces and present several series of such surfaces that are rigid with respect to the action of the deck transformation group.