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

### Abstract

A method of “algebraic estimates” is developed, and used to study the stability properties of integrals of the form $\int_B |f(z)|^{-\delta} dV$, under small deformations of the function $f$. The estimates are described in terms of a stratification of the space of functions $\{R(z) = |P(z)|^\varepsilon/|Q(z)|^\delta\}$ 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 $c\to \int_B |f(z,c)|^{-\delta} dV_1\cdots d V_n$ 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.

Duong H. Phong

Jacob Sturm