Resolution of singularities of the cotangent sheaf of a singular variety

Abstract

The main problem studied here is resolution of singularities of the cotangent sheaf of a complex- or real-analytic space X0 (or of an algebraic variety X0 over a field of characteristic zero). Given X0, we ask whether there is a global resolution of singularities σ:X→X0 such that the pulled-back cotangent sheaf of X0 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 σ. 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 X0; there have been important applications of the latter to L2-cohomology.