{
  "id": "math/0007202",
  "version": "v1",
  "published": "2000-07-01T00:00:00.000Z",
  "updated": "2000-07-01T00:00:00.000Z",
  "title": "Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions",
  "authors": [
    "D. H. Phong",
    "Jacob Sturm"
  ],
  "comment": "53 pages, published version",
  "journal": "Ann. of Math. (2) 152 (2000), no. 1, 277-329",
  "categories": [
    "math.NT"
  ],
  "abstract": "A method of ``algebraic estimates'' is developed, and used to study the stability properties of integrals of the form \\int_B|f(z)|^{-\\d}dV, under small deformations of the function f. The estimates are described in terms of a stratification of the space of functions \\{R(z)=|P(z)|^{\\e}/|Q(z)|^{\\d}\\} by algebraic varieties, on each of which the size of the integral of R(z) is given by an explicit algebraic expression. The method gives an independent proof of a result on stability of Tian in 2 dimensions, as well as a partial extension of this result to 3 dimensions. In arbitrary dimensions, combined with a key lemma of Siu, it establishes the continuity of the mapping c\\ra \\int_B|f(z,c)|^{-\\d}dV_1\\cdots dV_n when f(z,c) is a holomorphic function of (z,c). In particular the leading pole is semicontinuous in f, strengthening also an earlier result of Lichtin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-07-01T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local zeta functions",
      "algebraic estimates",
      "distribution functions",
      "uniform estimates",
      "explicit algebraic expression"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Princeton University and the Institute for Advanced Study",
      "journal": "Ann. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......7202P"
    }
  }
}