{
  "id": "2107.04841",
  "version": "v1",
  "published": "2021-07-10T14:07:08.000Z",
  "updated": "2021-07-10T14:07:08.000Z",
  "title": "The space of finite-energy metrics over a degeneration of complex manifolds",
  "authors": [
    "Rémi Reboulet"
  ],
  "comment": "48 pages",
  "categories": [
    "math.DG",
    "math.AG",
    "math.CV"
  ],
  "abstract": "Given a degeneration of compact projective complex manifolds $X$ over the punctured disc, with meromorphic singularities, and a relatively ample line bundle $L$ on $X$, we study spaces of plurisubharmonic metrics on $L$, with particular focus on (relative) finite-energy conditions. We endow the space $\\hat \\cE^1(L)$ of relatively maximal, relative finite-energy metrics with a $d_1$-type distance given by the Lelong number at zero of the collection of fibrewise Darvas $d_1$-distances. We show that this metric structure is complete and geodesic. Seeing $X$ and $L$ as schemes $X_\\K$, $L_\\K$ over the discretely-valued field $\\K=\\mathbb{C}((t))$ of complex Laurent series, we show that the space $\\cE^1(L_\\K\\an)$ of non-Archimedean finite-energy metrics over $L_\\K\\an$ embeds isometrically and geodesically into $\\hat \\cE^1(L)$, and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case. We investigate consequences regarding convexity of non-Archimedean functionals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-10T14:07:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "degeneration",
      "compact projective complex manifolds",
      "non-archimedean finite-energy metrics",
      "complex laurent series",
      "relatively ample line bundle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}