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

A method of "algebraic estimates" is developed, and used to. study the stability properties of integrals of the form integral (B)/f(z)/(-delta)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)/(delta)} 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 --> integral (B)/f(z, c)/(-delta)dV(1)...dV(n) when f(z,c) isa holomorphic function of (z,c). In particular the leading pole is semicontinuous in f, strengthening also an earlier result of Lichtin.