{
  "id": "2211.10592",
  "version": "v1",
  "published": "2022-11-19T05:38:25.000Z",
  "updated": "2022-11-19T05:38:25.000Z",
  "title": "Likely intersections",
  "authors": [
    "Sebastian Eterović",
    "Thomas Scanlon"
  ],
  "categories": [
    "math.AG",
    "math.LO",
    "math.NT"
  ],
  "abstract": "We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient spaces definable in some fixed o-minimal expansion of the ordered field of real numbers. For an instance of our general result, consider the case of subvarieties of Shimura varieties. Let $S$ be a Shimura variety. Let $\\pi:D \\to \\Gamma \\backslash D = S$ realize $S$ as a quotient of $D$, a homogeneous space for the action of a real algebraic group $G$, by the action of $\\Gamma < G$, an arithmetic subgroup. Let $S' \\subseteq S$ be a special subvariety of $S$ realized as $\\pi(D')$ for $D' \\subseteq D$ a homogeneous space for an algebraic subgroup of $G$. Let $X \\subseteq S$ be an irreducible subvariety of $S$ not contained in any proper weakly special subvariety of $S$. Assume that the intersection of $X$ with $S'$ is persistently likely meaning that whenever $\\zeta:S_1 \\to S$ and $\\xi:S_1 \\to S_2$ are maps of Shimura varieties (meaning regular maps of varieties induced by maps of the corresponding Shimura data) with $\\zeta$ finite, $\\dim \\xi \\zeta^{-1} X + \\dim \\xi \\zeta^{-1} S' \\geq \\dim \\xi S_1$. Then $X \\cap \\bigcup_{g \\in G, \\pi(g D') \\text{ is special }} \\pi(d D')$ is dense in $X$ for the Euclidean topology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-19T05:38:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C64",
      "11G18",
      "14D07",
      "14G35"
    ],
    "keywords": [
      "intersection",
      "shimura variety",
      "proper weakly special subvariety",
      "real algebraic group",
      "homogeneous space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}