Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions

D. H. Phong, Jacob Sturm

Research output: Contribution to journalArticle

37 Scopus citations

Abstract

A method of "algebraic estimates" is developed, and used to study the stability properties of integrals of the form fB|f(z)|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)|ε/|Q(z)|δ} 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 → fB |f(z, c)|dV1 ⋯ dVn 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.

Original languageEnglish (US)
Pages (from-to)277-329
Number of pages53
JournalAnnals of Mathematics
Volume152
Issue number1
DOIs
StatePublished - Jul 2000

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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