Polar loci of multivariable archimedean zeta functions

Abstract

We determine, up to exponentiating, the polar locus of the multivariable archimedean zeta function associated to a finite collection of polynomials F. The result is the monodromy support locus of F, a topological invariant. We give a relation between the multiplicities of the irreducible components of the monodromy support locus and the polar orders. These generalize results of Barlet for the case when F is a single polynomial. Our result determines the slopes of the polar locus of the zeta function of F, closing a circle of results of Loeser, Maisonobe, Sabbah. We apply our main result to elucidate the topological information contained by the oblique part of the zero locus of any ideal of Bernstein-Sato type.