{
  "id": "2405.03312",
  "version": "v1",
  "published": "2024-05-06T09:40:41.000Z",
  "updated": "2024-05-06T09:40:41.000Z",
  "title": "$Z$-critical equations for holomorphic vector bundles on Kähler surfaces",
  "authors": [
    "Julien Keller",
    "Carlo Scarpa"
  ],
  "comment": "38 pages. Comments are welcome!",
  "categories": [
    "math.DG",
    "math.AG"
  ],
  "abstract": "We prove that the existence of a $Z$-positive and $Z$-critical Hermitian metric on a rank 2 holomorphic vector bundle over a compact K\\\"ahler surface implies that the bundle is $Z$-stable. As particular cases, we obtain stability results for the deformed Hermitian Yang-Mills equation and the almost Hermite-Einstein equation for rank 2 bundles over surfaces. We show examples of $Z$-unstable bundles and $Z$-critical metrics away from the large volume limit.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-06T09:40:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C07",
      "14J60",
      "32W50"
    ],
    "keywords": [
      "holomorphic vector bundle",
      "kähler surfaces",
      "critical equations",
      "large volume limit",
      "deformed hermitian yang-mills equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}