{
  "id": "2109.05089",
  "version": "v1",
  "published": "2021-09-10T20:04:56.000Z",
  "updated": "2021-09-10T20:04:56.000Z",
  "title": "On the Thom conjecture in $CP^3$",
  "authors": [
    "Daniel Ruberman",
    "Marko Slapar",
    "Sašo Strle"
  ],
  "comment": "13 pages, 2 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "What is the simplest smooth simply connected 4-manifold embedded in $CP^3$ homologous to a degree $d$ hypersurface $V_d$? A version of this question associated with Thom asks if $V_d$ has the smallest $b_2$ among all such manifolds. While this is true for degree at most $4$, we show that for all $d \\geq 5$, there is a manifold $M_d$ in this homology class with $b_2(M_d) < b_2(V_d)$. This contrasts with the Kronheimer-Mrowka solution of the Thom conjecture about surfaces in $CP^2$, and is similar to results of Freedman for $2n$-manifolds in $CP^{n+1}$ with $n$ odd and greater than $1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-10T20:04:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K40",
      "57R40",
      "57R95",
      "14J70"
    ],
    "keywords": [
      "thom conjecture",
      "homology class",
      "simplest smooth",
      "kronheimer-mrowka solution",
      "thom asks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}