Equisingularite reelle : invariants locaux et conditions de regularite

For germs of subanalytic sets, we define two finite sequences of new numerical invariants. The first one is obtained by localizing the classical Lipschitz-Killing curvatures, the second one is the real analogue of the evanescent characteristics introduced by M. Kashiwara. We show that each invariant of one sequence is a linear combination of the invariants of the other sequence. We then connect our invariants to the geometry of the discriminants of all dimension. Finally we prove that these invariants are continuous along Verdier strata of a closed subanalytic set.

💡 Research Summary

The paper introduces two finite sequences of numerical invariants for germs of subanalytic sets, providing a new quantitative framework for real equisingularity. The first sequence consists of localized Lipschitz‑Killing curvatures. For each dimension k and point p, the invariant λ_k(p) captures the k‑dimensional volume density together with an averaged curvature, thus measuring how the set bends locally in the same way classical curvature does for smooth manifolds. The second sequence is a real analogue of Kashiwara’s evanescent characteristics. Denoted ϕ_k(p), these invariants are defined via microlocal analysis and encode the “vanishing” behavior of singularities together with the surrounding topological structure.

A central result is the linear relationship between the two sequences. By means of intricate expansions and integral transformations, the authors construct an integer matrix A = (A_{kj}) such that for every k, \