In this paper we initiate a study on the notion of G-minimality of four-manifolds equipped with an action of a finite group G. Our work shows that even in the case of cyclic actions on CP2#CP2, the comparison of G-minimality in the various categories (i.e., locally linear, smooth, symplectic) is already a delicate and interesting problem. In particular, we show that if a symplectic Zn-action on CP2#CP2 has an invariant locally linear topological (−1)-sphere, then it must admit an invariant symplectic (−1)-sphere, provided that n = 2 or n is odd. For the case where n > 2 and even, the same conclusion is true under a stronger assumption, i.e., the invariant (−1)-sphere is smoothly embedded. The techniques developed in this paper also find applications in the smooth classification of G-Hirzebruch surfaces. More precisely, we give a classification of G-Hirzebruch surfaces (each equipped with a homologically trivial, holomorphic G = Zn-action) up to orientation-preserving equivariant diffeomorphisms. The main technical issue encountered in the classifica- tion is to distinguish non-diffeomorphic G-Hirzebruch surfaces which have the same fixed-point set structure, and an interesting discovery of this paper is that a cer- tain “equivariant Gromov-Taubes invariant”, i.e., an invariant defined by counting certain embedded invariant negative two-spheres, can be used to distinguish such G-Hirzebruch surfaces. Finally, going back to the original question of G-minimality, we show that for G = Zn, a minimal rational G-surface is minimal as a symplectic G-manifold if and only if it is minimal as a smooth G-manifold.
Available at: http://works.bepress.com/weiminchen_chen/13/