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**Almost étale resolution of foliations.**
*(English)*
Zbl 1295.32041

The authors study the problem of resolution of singularities of complex foliated varieties.

This problem was solved by A. Seidenberg in dimension \(2\) [Am. J. Math. 90, 248–269 (1968; Zbl 0159.33303)] and by F. Cano for codimension \(1\) foliations in dimension \(3\) [Ann. Math. (2) 160, No. 3, 907–1011 (2005; Zbl 1088.32019)]. However, already the case of \(3\)-folds foliated by curves is much more complicated, as such \(3\)-folds need not have resolutions with smooth base space and canonical singularities (this fact is stated as a proposition on page 280 but as with the remaining theorems in the introduction it is difficult to find the corresponding statement in the main body of the paper).

The main theorem of the paper says that in order to solve the resolution problem for \(3\)-folds foliated by curves one should pass to complex Deligne-Mumford stacks (in the paper: “champs”) and apply the methods used by the second author [Acta Math. 197, No. 2, 167–289 (2006; Zbl 1112.37016)] to solve the resolution problem in case of real manifolds with boundary. More precisely (but still roughly speaking), the authors prove that for any foliated \(3\)-dimensional “champ” in complex spaces with boundary one can find its resolution which has canonical singularities and normal crossing boundary, using a sequence of either smoothed weighted blow ups or certain modifications in invariant centres (in the \(2\)-category of smooth logarithmic Deligne-Mumford “champs”). This theorem plays an important role in the minimal model program for foliations by curves, which is developed in a series of preprints by the first author.

This problem was solved by A. Seidenberg in dimension \(2\) [Am. J. Math. 90, 248–269 (1968; Zbl 0159.33303)] and by F. Cano for codimension \(1\) foliations in dimension \(3\) [Ann. Math. (2) 160, No. 3, 907–1011 (2005; Zbl 1088.32019)]. However, already the case of \(3\)-folds foliated by curves is much more complicated, as such \(3\)-folds need not have resolutions with smooth base space and canonical singularities (this fact is stated as a proposition on page 280 but as with the remaining theorems in the introduction it is difficult to find the corresponding statement in the main body of the paper).

The main theorem of the paper says that in order to solve the resolution problem for \(3\)-folds foliated by curves one should pass to complex Deligne-Mumford stacks (in the paper: “champs”) and apply the methods used by the second author [Acta Math. 197, No. 2, 167–289 (2006; Zbl 1112.37016)] to solve the resolution problem in case of real manifolds with boundary. More precisely (but still roughly speaking), the authors prove that for any foliated \(3\)-dimensional “champ” in complex spaces with boundary one can find its resolution which has canonical singularities and normal crossing boundary, using a sequence of either smoothed weighted blow ups or certain modifications in invariant centres (in the \(2\)-category of smooth logarithmic Deligne-Mumford “champs”). This theorem plays an important role in the minimal model program for foliations by curves, which is developed in a series of preprints by the first author.

Reviewer: Adrian Langer (Warszawa)