Singularites reelles isolees et developpements asymptotiques d'integrales oscillantes

Abstract

Let (X_R, 0) be a germ of real analytic subset in (R^N, 0) of pure dimension n+1 with an isolated singularity at 0. Let (f_R,0) : (X_R, 0) --> (R,0) a real analytic germ with an isolated singularity at 0, such that its complexification f_C vanishes on the singular set S of X_C. We also assume that X_R-[0] is orientable. To each $A\in H^0(X_{\mathbb{R}}-\{0\},\mathbb{C})$ we associate a $n-$cycle $\Gamma (A)$ ("explicitly " described) in the complex Milnor fiber of $f_{\mathbb{C}}$ at 0 such that the non trivial terms in the asymptotic expansions of the oscillating integrals ${\int }_A{e}^{i\tau f(x)}\varphi (x)$ when $\tau \to \pm \infty$ can be read from the spectral decomposition of $\Gamma (A)$ relative to the monodromy of $f_{\mathbb{C}}$ at 0 .