Oblique poles of $\int_X| {f}| ^{2\lambda}| {g}|^{2\mu} \square$

TL;DR

This paper investigates the existence of oblique polar lines in the meromorphic extension of a current-valued integral involving holomorphic functions, using monodromy and stratification techniques under specific geometric conditions.

Contribution

It establishes conditions for oblique polar lines in the meromorphic extension of integrals involving holomorphic germs, introducing new methods involving monodromy and stratification.

Findings

Oblique polar lines exist under specified geometric conditions.

Monodromy of local systems influences the extension.

Two detailed examples demonstrate the theory.

Abstract

Existence of oblique polar lines for the meromorphic extension of the current valued function $\int |f|^{2\lambda}|g|^{2\mu}\square$ is given under the following hypotheses: $f$ and $g$ are holomorphic function germs in $\CC^{n+1}$ such that $g$ is non-singular, the germ $S:=\ens{\d f\wedge \d g =0}$ is one dimensional, and $g|_S$ is proper and finite. The main tools we use are interaction of strata for $f$ (see \cite{B:91}), monodromy of the local system $H^{n-1}(u)$ on $S$ for a given eigenvalue $\exp(-2i\pi u)$ of the monodromy of $f$, and the monodromy of the cover $g|_S$. Two non-trivial examples are completely worked out.