Oblique Polar Lines of ∫ _X |f|^2λ|g|^2μ□

Abstract

Existence of oblique polar lines for the meromorphic extension of the current valued function ∫|f|^2λ|g|^2μ□ is given under the following hypotheses: f and g are holomorphic function germs in ℂⁿ⁺¹ such that g is non-singular, the germ Σ := {df ∧ dg = 0} is one dimensional, and g is proper and finite on S := {df = 0}. The main tools we use are interaction of strata for f (see [4]), monodromy of the local system H ^n-1 (u) on S for a given eigenvalue exp(−2iπu) of the monodromy of f, and the monodromy of the cover gS. Two non-trivial examples are completely worked out.