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

Daniel Barlet (1, 2) , H.-M. Maire (3)

1 IECN - Institut Élie Cartan de Nancy
2 IUF - Institut universitaire de France
3 UNIGE - Université de Genève = University of Geneva

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.

Mots clés: asymptotic expansion; fibre integrals; meromorphic extension

Domaines: Géométrie algébrique [math.AG]

Daniel Barlet, H.-M. Maire. Oblique poles of \ $\int_X\vert {f}\vert ^{2\lambda}\vert {g}\vert^{2\mu}\ \square$. Complex Analysis, Ebenfelt P. and others Jul 2008, Fribourg, Switzerland. p.1-23. ⟨hal-00354565⟩