Asymptotic equisingularity and topology of complex hypersurfaces

Abstract

We consider an equisingularity problem for polynomial families of affine hypersurfaces $X_τ\subset \mathbb C^n$ with (at worst) isolated singularities. We show that the constancy of the global polar invariants $γ^* (X_τ)$ is equivalent to the $t$-equisingularity at infinity, an asymptotic-type equisingularity that we introduce. We prove that $γ^*$-constancy implies C$^\infty$-triviality in the neighbourhood of infinity. We show how the invariants $γ^*$ enter in the description of a CW-complex model of a hypersurface $X_τ$ and therefore provide in particular new invariants at infinity for polynomial functions $f: \mathbb C^n \to \mathbb C$.