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 ∫ B |f(z)| −δdV , 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→ ∫ B |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.