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

D. H. Phong, Jacob Sturm

Published 2000-07-01Version 1

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.