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 ${\gamma }^{\ast }(X_{\tau })$ is equivalent to the $t$-equisingularity at infinity, an asymptotic-type equisingularity that we introduce. We prove that ${\gamma }^{\ast }$-constancy implies C${}^{\infty }$-triviality in the neighbourhood of infinity. We show how the invariants ${\gamma }^{\ast }$ 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 $f:{\mathbb{C}}^n\to \mathbb{C}$.

May 1997, revised Oct. 1997, changed title, last version before publication

Cited by in corpus (14)

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• Polar degree and vanishing cycles
• Bifurcation values of polynomial functions and perverse sheaves
• Global Euler obstruction, global Brasselet numbers and critical points
• Detecting asymptotic non-regular values by polar curves
• Arrangements, Milnor Fibers and Polar Curves
• Vanishing cycles of pencils of hypersurfaces
• Index formula for MacPherson cycles of affine algebraic varieties
• Complements of hypersurfaces, variation maps and minimal models of arrangements
• Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements