Real canonical cycle and asymptotics of oscillating integrals

Abstract

Let $X_{\mathbb R} \subset {\mathbb R}^{N}$ a real analytic set such that its complexification $X_{\mathbb C} \subset {\mathbb C}^{N}$ is normal with an isolated singularity at $0$. Let $f_{\mathbb R} : X_{\mathbb R} \rightarrow {\mathbb R}$ a real analytic function such that its complexification $f_{\mathbb C} : X_{\mathbb C} \rightarrow {\mathbb C}$ has an isolated singularity at $0$ in $X_{\mathbb C}$. Assuming an orientation given on $X_{\mathbb R}^{*}$, to a connected component $A$ of $X_{\mathbb R}^{*}$ we associate a compact cycle $\Gamma (A)$ in the Milnor fiber of $f_{\mathbb C}$ which determines completely the poles of the meromorphic extension of $\int_{A} f^{\lambda} \square$ or equivalently the asymptotics when $\tau \rightarrow \pm \infty$ of the oscillating integrals $\int_{A} e^{i \tau f} \square$. A topological construction of $\Gamma(A)$ is given. This completes the results of [BM] paragraph 6.