{
  "id": "2303.09809",
  "version": "v1",
  "published": "2023-03-17T07:33:49.000Z",
  "updated": "2023-03-17T07:33:49.000Z",
  "title": "On tropical Hodge theory for tropical varieties",
  "authors": [
    "Ryota Mikami"
  ],
  "comment": "35 pages",
  "categories": [
    "math.AG",
    "math.CO",
    "math.DG"
  ],
  "abstract": "To prove log-concavity of the characteristic polynomials of matroids, Adiprasito-Huh-Katz proved the K\\\"{a}hler package (the hard Lefschetz theorem and the Hodge-Riemann bilinear relations) for their Chow rings. Amini-Piquerez generalized it to tropical cohomology of smooth projective tropical varieties. Their proofs were combinatorial. In this paper, we establish tropical Hodge theory except for regularity of solutions of Laplacian, and give a conditional proof of the K\\\"{a}hler package in the same way as compact K\\\"{a}hler manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-17T07:33:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14T20"
    ],
    "keywords": [
      "hodge-riemann bilinear relations",
      "hard lefschetz theorem",
      "characteristic polynomials",
      "chow rings",
      "smooth projective tropical varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}