Isolated real singularities and asymptotic expansions for oscillating integrals. (Singularités réelles isolées et développements asymptotiques d’intégrales oscillantes.) (French) Zbl 1080.32028

Guillopé, L. (ed.) et al., Proceedings of the colloquium dedicated to the memory of Jean Leray, Nantes, France, June 17–18, 2002. Paris: Société Mathématique de France (ISBN 2-85629-160-0/pbk). Séminaires et Congrès 9, 25-50 (2004).
Summary: Let \((X_\mathbb{R},0)\) be a germ of real analytic subset in \((\mathbb{R}^N,0)\) of pure dimension \(n+1\) with an isolated singularity at 0. Let \[ (f_\mathbb{R},0): (X_\mathbb{R},0)\to(\mathbb{R},0) \] a real analytic germ with an isolated singularity at 0, such that its complexification \(f_\mathbb{C}\) vanishes on the singular set \(S\) of \(X_\mathbb{C}\). We also assume that \(X_\mathbb{R} \setminus\{0\}\) is orientable. To each \(A\in H^0(X_\mathbb{R}\setminus \{0\}, \mathbb{C})\) we associate an \(n\)-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 \(\int_Ae^{i\tau f(x)}\varphi(x)\) 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.
For the entire collection see [Zbl 1056.00013].


32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
32C30 Integration on analytic sets and spaces, currents