Functorial resolution except for toroidal locus. Toroidal compactification

TL;DR

This paper develops a canonical resolution method for algebraic varieties in characteristic zero that preserves a specified open subset with toroidal singularities, generalizing Hironaka's desingularization and enabling toroidal compactifications.

Contribution

It introduces a functorial desingularization process that preserves a toroidal open subset, extending previous methods to more general varieties and singularities.

Findings

Existence of a canonical desingularization that preserves the toroidal locus.

Generalization of Hironaka's desingularization to varieties with toroidal singularities.

Application to toroidal equisingular compactifications.

Abstract

Let $X$ be any variety in characteristic zero. Let $V \subset X$ 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 $f: Y \to X$ , 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-Wlodarczyk. 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.