Some applications of microlocalization for local complete intersection subvarieties

TL;DR

This paper extends Saito's microlocalization to local complete intersection subvarieties, establishing key properties of the minimal exponent and its relation to Bernstein-Sato roots and spectral numbers, with applications to stratification and equisingularity.

Contribution

It introduces a microlocalization analogue for LCI subvarieties and proves its relation to existing invariants, advancing understanding of singularity invariants in this context.

Findings

Minimal exponent equals the smallest Bernstein-Sato root.

In ICIS, minimal exponent matches the smallest non-zero spectral number.

Minimal exponent is constant in certain equisingular families.

Abstract

Saito's microlocalization construction has been used to great effect in understanding hypersurface singularities. In this paper, we introduce what we believe to be a suitable analogue of the microlocalization construction for local complete intersection subvarieties. As evidence, we relate our construction to Saito's in the codimension one case. Moreover, we use this construction to study various natural questions concerning the minimal exponent of LCI subvarieties. We show that the minimal exponent agrees with the smallest Bernstein-Sato root, which was expected to be true. We also show that, in the isolated complete intersection singularities case, the minimal exponent agrees with the smallest non-zero spectral number, as defined by Dimca, Maisonobe and Saito. As applications of these results, we prove constructibility of the minimal exponent along certain Whitney stratifications and we prove that the spectrum (hence, the minimal exponent) is constant in equisingular families of ICIS varieties, in the sense of Gaffney.