Curvature integrals on the real Milnor fibre

Nicolas Dutertre

doi:10.4171/cmh/124

JournalscmhVol. 83, No. 2pp. 241–288

Curvature integrals on the real Milnor fibre

Nicolas Dutertre
Université de Provence, Marseille, France

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Abstract

Let $f : R^{n + 1} \to R$ be a polynomial with an isolated critical point at $0$ and let $f_{t} : R^{n + 1} \to R$ be a one-parameter deformation of $f$ . We study the differential geometry of the real Milnor fiber $C_{t}^{ε} = f_{t}^{- 1} (0) \cap B_{ε}^{n + 1}$ . More precisely, we express the limits

ε \to 0 lim t \to 0 lim 1 ε^{k} \int_{C_{t}^{ε}} s_{n - k} (x) d x,

where $s_{n - k}$ is the $(n - k)$ th symmetric function of curvature, in terms of the following averages of topological degrees:

\int_{G_{n + 1}^{k}} de g_{0} \nabla (f ∣_{H}) d H,

where $G_{n + 1}^{k}$ is the Grassmann manifold of $k$ -dimensional planes through the origin of $R^{n + 1}$ .

When $0$ is an algebraically isolated critical point, we study the limits

ε \to 0 lim t \to 0 lim 1 ε^{k} \int_{C_{t}^{ε}} h_{n - k} (x) d x,

where the $h_{n - k}$ are positive extrinsic curvature functions. We prove that these limits are finite and that they are bounded in terms of the Milnor–Teissier numbers of the complexification of $f$ .

Cite this article

Nicolas Dutertre, Curvature integrals on the real Milnor fibre. Comment. Math. Helv. 83 (2008), no. 2, pp. 241–288

DOI 10.4171/CMH/124