Deformation Formulas for Parameterizable Hypersurfaces

We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. Assuming isolated polar activity at the origin, we fully express the Lê numbers of the special fiber in terms of those of the generic fiber and the characteristic polar multiplicities of the comparison complex—a perverse sheaf naturally associated to any reduced complex analytic space on which the shifted constant sheaf \( \mathbb{Q}_X[\dim X] \) is perverse. This generalizes the classical Milnor number formula for plane curves in terms of double points and also recovers results of Gaffney, Bobadilla, and Mond. We extend the method to maps from \( \mathbb{C}^2 \to \mathbb{C}^3 \), and propose an ansatz for a deformation formula applicable across Mather’s nice dimensions.

Annales de l’Institut Fourier, Volume 74, No. 3, pp. 1153–1188