{
  "id": "2401.02010",
  "version": "v1",
  "published": "2024-01-04T00:42:40.000Z",
  "updated": "2024-01-04T00:42:40.000Z",
  "title": "Semistability of pairs for projective toric varieties",
  "authors": [
    "Naoto Yotsutani"
  ],
  "comment": "27 pages, no figure. Comments are welcome",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X \\to \\mathbb P^N$ be a smooth linearly normal projective variety. It was proved by Paul that the $K$-energy of $(X,\\omega_{FS}|_{X})$ restricted to the Bergman metrics is bounded from below if and only if the pair of (rescaled) Chow/Hurwitz forms of $X$ is semistable. In this paper, we provide a necessary and sufficient condition for a given smooth toric variety $X_P$ to be semistable of pairs with respect to $\\mathcal O_{X_P}(i)$ for a positive integer $i$. Applying this result to a smooth polarized toric variety $(X_P, L_P)$, we prove that $(X_P, L_P)$ is asymptotically semistable of pairs if and only if it is K-semistable for toric degenerations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-04T00:42:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51M20",
      "14M25",
      "53C55"
    ],
    "keywords": [
      "projective toric varieties",
      "semistability",
      "smooth polarized toric variety",
      "smooth linearly normal projective variety",
      "smooth toric variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}