Deformation Formulas for Parameterized Hypersurfaces

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 Lê numbers of the special fiber in terms of the Lê 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 [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.

The Weight Filtration on the Constant Sheaf on a Parameterized Space

“…X is called the comparison complex on X, and was first defined by the author and David Massey in [9] and subsequently studied in several papers by the author [7], [8] and Massey [13].…”

“…We have called such spaces X with Q-homology manifold normalizations parameterized spaces in [9], [7], and [8].…”

“…Taking the alternating sums of the dimensions of the terms in the resulting short exact sequence (7) 0…”

“…Consider the short exact sequence (7) at the end of the proof of Theorem 5.1; in particular, the middle term C ker{id −h C }.…”

Rational homology manifolds and hypersurface normalizations

“…We refer to this exact sequence as the fundamental short exact sequence of the normalization. This short exact sequence, and the perverse sheaf N • X in particular, have been examined recently in several papers by the author and D. Massey in the case where the normalization Y is smooth ( [4], [3]), where N • X is called the multiple-point complex of the normalization (see Section 2).…”