Deformation Formulas for Parameterized Hypersurfaces

TL;DR

This paper derives formulas relating the topological invariants of hypersurfaces under one-parameter deformations, generalizing classical results and applying to complex analytic spaces and mappings.

Contribution

It provides a comprehensive framework to express Lê numbers of deformed hypersurfaces in terms of generic fibers and characteristic polar multiplicities, extending classical deformation formulas.

Findings

Expressed Lê numbers of special fibers via generic fibers and polar multiplicities.

Generalized Milnor number formulas for hypersurfaces and maps.

Connected deformation formulas to existing results of Gaffney and Bobadilla.

Abstract

We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the L\^{e} numbers of the special fiber in terms of the L\^{e} numbers of the generic fiber and the characteristic polar multiplicities of the comparison, a perverse sheaf naturally associated to any reduced complex analytic space on which the constant sheaf $\Q_X^\bullet[\dim X]$ is perverse. This generalizes the classical formula for the Milnor number of a plane curve in terms of double points as well as Mond's image Milnor number. We also recover results of Gaffney and Bobadilla using this framework. We obtain similar deformation formulas for maps from $\C^2$ to $\C^3$, and provide an ansatz for obtaining deformation formulas for all dimensions within Mather's nice dimensions.