{
  "id": "math/0606423",
  "version": "v2",
  "published": "2006-06-17T17:32:12.000Z",
  "updated": "2006-06-30T00:58:08.000Z",
  "title": "Test Configurations for K-Stability and Geodesic Rays",
  "authors": [
    "D. H. Phong",
    "Jacob Sturm"
  ],
  "comment": "27 pages, no figure; references added; typos corrected",
  "categories": [
    "math.DG",
    "math.AG"
  ],
  "abstract": "Let $X$ be a compact complex manifold, $L\\to X$ an ample line bundle over $X$, and ${\\cal H}$ the space of all positively curved metrics on $L$. We show that a pair $(h_0,T)$ consisting of a point $h_0\\in {\\cal H}$ and a test configuration $T=({\\cal L}\\to {\\cal X}\\to {\\bf C})$, canonically determines a weak geodesic ray $R(h_0,T)$ in ${\\cal H}$ which emanates from $h_0$. Thus a test configuration behaves like a vector field on the space of K\\\"ahler potentials ${\\cal H}$. We prove that $R$ is non-trivial if the ${\\bf C}^\\times$ action on $X_0$, the central fiber of $\\cal X$, is non-trivial. The ray $R$ is obtained as limit of smooth geodesic rays $R_k\\subset{\\cal H}_k$, where ${\\cal H}_k\\subset{\\cal H}$ is the subspace of Bergman metrics.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-06-30T00:58:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "k-stability",
      "test configuration behaves",
      "ample line bundle",
      "weak geodesic ray",
      "compact complex manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......6423P"
    }
  }
}