Symplectic rational $G$-surfaces and equivariant symplectic cones

Abstract

We give characterizations of a finite group $G$ acting symplectically on a rational surface ($\mathbb{CP}^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surfaces for the corresponding $G$-rational surfaces, analogous to a classical result in algebraic geometry. Besides the characterizations of the group $G$ (which is completely determined for the case of $\mathbb{CP}^2 \# N \overline{\mathbb{CP}^2}, N={2,3,4}$), we also investigate the equivariant symplectic minimality and equivariant symplectic cone of a given $G$-rational surface.