{
  "id": "1312.0848",
  "version": "v3",
  "published": "2013-12-03T15:19:41.000Z",
  "updated": "2015-03-08T04:27:45.000Z",
  "title": "$G$-minimality and invariant negative spheres in $G$-Hirzebruch surfaces",
  "authors": [
    "Weimin Chen"
  ],
  "comment": "36 pages, no figures, Journal of Topology",
  "categories": [
    "math.GT"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-01-13T22:47:00.000Z",
      "abstract": "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 $CP^2\\# \\overline{CP^2}$, 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 $Z_n$-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 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 $G=Z_n$-action) up to orientation-preserving equivariant diffeomorphisms. The main technical issue encountered in the classification 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 certain \"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 $G=Z_n$, a minimal rational $G$-surface is minimal as a symplectic $G$-manifold if and only if it is minimal as a smooth $G$-manifold.",
      "comment": "33 pages, no figures, title changed, references added, some minor errors corrected, submitted",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-03-08T04:27:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R57",
      "57S17",
      "57R17"
    ],
    "keywords": [
      "hirzebruch surfaces",
      "invariant negative spheres",
      "minimality",
      "symplectic",
      "equivariant gromov-taubes invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.0848C"
    }
  }
}