{
  "id": "2012.05416",
  "version": "v2",
  "published": "2020-12-10T02:25:12.000Z",
  "updated": "2022-08-30T18:20:35.000Z",
  "title": "The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces",
  "authors": [
    "Tristan C. Collins",
    "Adam Jacob",
    "Yu-Shen Lin"
  ],
  "comment": "v2: some improvements, 74 pages",
  "categories": [
    "math.DG",
    "math.AG"
  ],
  "abstract": "We prove the Strominger-Yau-Zaslow mirror symmetry conjecture for non-compact Calabi-Yau surfaces arising from, on the one hand, pairs $(\\check{Y},\\check{D})$ of a del Pezzo surface $\\check{Y}$ and $\\check{D}$ a smooth anti-canonical divisor and, on the other hand, pairs $(Y,D)$ of a rational elliptic surface $Y$, and $D$ a singular fiber of Kodaira type $I_k$. Three main results are established concerning the latter pairs $(Y,D)$. First, adapting work of Hein, we prove the existence of a complete Calabi-Yau metric on $Y\\setminus D$ asymptotic to a (generically non-standard) semi-flat metric in every K\\\"ahler class. Secondly, we prove an optimal uniqueness theorem to the effect that, modulo automorphisms, every K\\\"ahler class on $Y\\setminus D$ admits a unique asymptotically semi-flat Calabi-Yau metric. This result yields a finite dimensional K\\\"ahler moduli space of Calabi-Yau metrics on $Y\\setminus D$. Further, this result answers a question of Tian-Yau and settles a folklore conjecture of Yau in this setting. Thirdly, we prove that $Y\\setminus D$ equipped with an asymptotically semi-flat Calabi-Yau metric $\\omega_{CY}$ admits a special Lagrangian fibration whenever the de Rham cohomology class of $\\omega_{CY}$ is not topologically obstructed. Combining these results we define a mirror map from the moduli space of del Pezzo pairs $(\\check{Y}, \\check{D})$ to the complexified K\\\"ahler moduli of $(Y,D)$ and prove that the special Lagrangian fibration on $(Y,D)$ is $T$-dual to the special Lagrangian fibration on $(\\check{Y}, \\check{D})$ previously constructed by the authors. We give some applications of these results, including to the study of automorphisms of del Pezzo surfaces fixing an anti-canonical divisor.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-08-30T18:20:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "del pezzo surface",
      "syz mirror symmetry conjecture",
      "rational elliptic surface",
      "asymptotically semi-flat calabi-yau metric",
      "special lagrangian fibration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 74,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}