Functorial resolution except for toroidal locus. Toroidal compactification

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