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

By Duong H. Phong, Jacob Sturm. 2000

92 citations recorded by the source corpus.

A method of \algebraic estimates" is developed, and used to study the stability properties of integrals of the form R B jf(z)j i‐ dV , under small deformations of the function f. The estimates are described in terms of a stratiflcation of the space of functionsfR(z )= jP (z)j " =jQ(z)j ‐ g 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! R B jf(z;c)j i‐ 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4d8c487f11fc13138154077dd5220ead03f0b8d8

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Kähler currents and null loci (2013, background): “In turn, these were motivated by the seminal work of Phong, Stein and Sturm [35–37].”
Multilinear level set operators, oscillatory integral operators, and Newton polyhedra (2001, background): “Under this form,stability has been investigated by Karpushkin [5],Tian [18],Siu [14–15],and by the authors in [10] and [ 12 ]; • More globally,we may also seek uniform bounds for the above invariants which hold for all ˜”
A multi-dimensional resolution of singularities with applications to analysis (2010, unknown): “1”
Oscillatory integral decay, sublevel set growth, and the Newton polyhedron (2008, methodology): “In [ PSt1 ] and [PSt2], the method of algebraic estimates is used for this purpose; in [PSt2] resolution of singularities algorithms of Bierstone and Milman such as [BM] are also used.”
Oscillatory and Fourier integral operators with degenerate canonical relations (2002, background): “Related is the work by Phong, Stein and Sturm ([60], [62], [61]), with important contributions concerning the stability of estimates.”