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$G$-minimality and invariant negative spheres in $G$-Hirzebruch surfaces

In this paper a study of $G$-minimality, i.e., minimality of four-manifolds equipped with an action of a finite group $G$, is initiated. We focus on cyclic actions on $CP^2\# \overline{CP^2}$, and our work shows that even in this simple setting, the comparison of $G$-minimality in the various categories, i.e., locally linear, smooth, and symplectic, is already delicate and interesting. For example, we show that if a symplectic $Z_n$-action on $CP^2\# \overline{CP^2}$ 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 holds under a stronger assumption, i.e., the invariant $(-1)$-sphere is smoothly embedded. Along the way of these proofs we develop certain techniques for producing embedded invariant $J$-holomorphic two-spheres of self-intersection $-r$ under a weaker assumption of an invariant smooth $(-r)$-sphere for $r$ relatively small compared with the group order $n$. We then apply the techniques to give a classification of $G$-Hirzebruch surfaces (i.e., Hirzebruch surfaces equipped with a homologically trivial, holomorphic $G=Z_n$-action) up to orientation-preserving equivariant diffeomorphisms. The main issue of the classification is to distinguish non-diffeomorphic $G$-Hirzebruch surfaces which have the same fixed-point set structure. An interesting discovery is that these non-diffeomorphic $G$-Hirzebruch surfaces have distinct equivariant Gromov-Taubes invariant, giving the first examples of such kind. Going back to the original question of $G$-minimality, we show that for $G=Z_n$, a minimal rational $G$-surface is minimal as a symplectic $G$-manifold if and only if it is minimal as a smooth $G$-manifold.