Functorial resolution except for toroidal locus. Toroidal compactification

Abstract

Let X be any variety in characteristic zero. Let V⊂X be an open subset that has toroidal singularities. We show the existence of a canonical desingularization of X except for V. It is a morphism f:Y→X, which does not modify the subset V and transforms X into a toroidal embedding Y, with singularities extending those on V. Moreover, the exceptional divisor has simple normal crossings on Y.The theorem naturally generalizes the Hironaka canonical desingularization. It does not modify the nonsingular locus V and transforms X into a nonsingular variety Y.The proof uses, in particular, the canonical desingularization of logarithmic varieties recently proved by Abramovich-Temkin-Włodarczyk. It also relies on the established here canonical functorial desingularization of locally toric varieties with an unmodified open toroidal subset. As an application, we show the existence of a toroidal equisingular compactification of toroidal varieties. All the results here can be linked to a simple functorial combinatorial desingularization algorithm developed in this paper.