Dh. Phong et J. Sturm, Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions, ANN MATH, 152(1), 2000, pp. 277-329
A method of "algebraic estimates" is developed, and used to. study the stab
ility properties of integrals of the form integral (B)/f(z)/(-delta)dV, und
er small deformations of the function f. The estimates are described in ter
ms of a stratification of the space of functions {R(z) = /P(z)/(E)//Q(z)/(d
elta)} 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 ind
ependent 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 dimension
s, combined with a key lemma of Siu, it establishes the continuity of the m
apping c --> integral (B)/f(z, c)/(-delta)dV(1)...dV(n) when f(z,c) isa hol
omorphic function of (z,c). In particular the leading pole is semicontinuou
s in f, strengthening also an earlier result of Lichtin.