Symplectic rational \(G\)-surfaces and equivariant symplectic cones (Q2073261)

This paper is concerned with finite group actions by symplectomorphisms on symplectic rational surfaces. Such a surface is called a symplectic rational \(G\)-surface where \(G\) stands for some finite group. A rational \(G\)-surface is called minimal if there is no invariant union of disjoint \((-1)\)-spheres. The authors obtained a classification of minimal symplectic rational \(G\)-surfaces in the paper. In the setting of algebraic geometry, a rational \(G\)-surface is a rational surface with a finite group action by algebraic automorphisms. See the classical reference [\textit{Yu. I. Manin}, Sov. Math., Dokl. 8, 803--806 (1967; Zbl 0171.41603); translation from Dokl. Akad. Nauk SSSR 175, 28--30 (1967)] and the more recent reference [\textit{I. V. Dolgachev} and \textit{V. A. Iskovskikh}, Prog. Math. 269, 443--548 (2009; Zbl 1219.14015)] for classification of rational \(G\)-surface in the algebraic category. The work in the paper under review shows that a large part of the classical story of classification of complex rational \(G\)-surfaces holds in the symplectic setting as well. The symplectic setting is more general because any algebraic automorphism is a symplectomorphism for a \(G\)-invariant Kähler form. The authors show that tools from \(4\)-manifold topology and symplectic geometry can replace most part of the algebro-geometric arguments. They also obtained some information which seems to be more precise than the classical algebraic results. The classification is a little bit long to state here, we refer the reader to the paper itself for the statements of the theorems.