Local volumes, equisingularity, and generalized smoothability

TL;DR

This paper introduces a new invariant called restricted local volume to study equisingularity and smoothability of complex analytic singularities, providing numerical criteria and deformation results for various classes of singularities.

Contribution

It defines the restricted local volume as an invariant of equisingularity, characterizes vanishing of local volume for deficient conormal singularities, and establishes deformation results for multiple classes of singularities.

Findings

Restricted local volume controls Whitney-Thom equisingularity.

Vanishing local volume characterizes deficient conormal singularities.

Smoothable singularities deform to deficient conormal singularities.

Abstract

We introduce the restricted local volume of a relatively very ample invertible sheaf as an invariant of equisingularity by determining its change across families. We apply this result to give numerical control of Whitney-Thom (differential) equisingularity for families of isolated complex analytic singularities. The characterization of the vanishing of the local volume gives rise to the class of deficient conormal (dc) singularities. We introduce a notion of generalized smoothability by considering the class of singularities that deform to dc singularities. Using Whitney stratifications and the functoriality properties of conormal spaces we show that fibers of conormal spaces are well-behaved under transverse maps. Then by Thom's transversality, the structure theorems of Hilbert-Burch and Buchsbaum-Eisenbud, we show that all smoothable singularities of dimension at least 2, Cohen-Macaulay codimension 2, Gorenstein codimension 3, and more generally determinantal and Pfaffian singularities deform to dc singularities.