Symplectic rational $G$-surfaces and equivariant symplectic cones

TL;DR

This paper characterizes finite groups acting symplectically on rational surfaces, establishing a symplectic analogue of classical algebraic geometry results, and explores equivariant symplectic structures and minimality.

Contribution

It provides a symplectic classification of finite group actions on rational surfaces and analyzes equivariant symplectic cones and minimality properties.

Findings

Complete group characterizations for certain rational surfaces.

Symplectic dichotomy between G-conic bundles and G-del Pezzo surfaces.

Analysis of equivariant symplectic minimality and cones.

Abstract

We give characterizations of a finite group $G$ acting symplectically on a rational surface ($\mathbb{C}P^2$ 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 $\mathbb{C}P^2\# N\overline{\mathbb{C}P^2}$, $N=2,3,4$), we also investigate the equivariant symplectic minimality and equivariant symplectic cone of a given $G$-rational surface.