Resolution of singularities of the cotangent sheaf of a singular variety

TL;DR

This paper proves the resolution of singularities of the cotangent sheaf for complex or real-analytic varieties up to dimension three, enabling monomialization of associated metrics and advancing geometric analysis.

Contribution

It establishes the resolution of singularities of the cotangent sheaf in dimensions up to three, extending previous results beyond isolated surface singularities.

Findings

Resolution proven for dimensions up to three.

Monomialization of the Fitting ideals achieved.

Applications to metric monomialization and L2-cohomology.

Abstract

The main problem studied is resolution of singularities of the cotangent sheaf of a complex- or real-analytic variety Y (or of an algebraic variety Y over a field of characteristic zero). Given Y, we ask whether there is a global resolution of singularities s: X -> Y such that the pulled-back cotangent sheaf of Y is generated by differential monomials in suitable coordinates at every point of X ("Hsiang-Pati coordinates''). Desingularization of the cotangent sheaf is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of s. We prove resolution of singularities of the cotangent sheaf in dimension up to three. It was previously known for surfaces with isolated singularities (Hsiang-Pati 1985, Pardon-Stern 2001). Consequences include monomialization of the induced Fubini-Study metric on the smooth part of a complex projective variety Y; there have been important applications of the latter to L2-cohomology.