Asymptotic equisingularity and topology of complex hypersurfaces

Abstract

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