Abstract
Let M be a complex- or real-analytic manifold, \(\theta \) be a singular distribution and \(\mathcal {I}\) a coherent ideal sheaf defined on M. We prove the existence of a local resolution of singularities of \(\mathcal {I}\) that preserves the class of singularities of \(\theta \), under the hypothesis that the considered class of singularities is invariant by \(\theta \)-admissible blowings-up. In particular, if \(\theta \) is monomial, we prove the existence of a local resolution of singularities of \(\mathcal {I}\) that preserves the monomiality of the singular distribution \(\theta \).
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References
Belotto da Silva, A.: Resolution of singularities in foliated spaces. Ph.D. thesis. Université de Haute-Alsace, France (2013)
Belotto, A.: Global resolution of singularities subordinated to a \(1\)-dimensional foliation. J. Algebra 447, 397–423 (2016)
Belotto da Silva, A.: Local monomialization of a system of first integrals, preprint. arXiv:1411.5333 [math.CV] (2015)
Belotto da Silva, A., Bierstone, E., Grandjean, V., Milman, P.: Resolution of singularities of the cotangent sheaf of a singular variety, preprint. arXiv:1504.07280 [math.AG] (2015)
Bierstone, E., Milman, P.D.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128, 207–302 (1997)
Bierstone, E., Milman, P.D.: Functoriality in resolution of singularities. Publ. Res. Inst. Math. Sci. 44(2), 609–639 (2008)
Cano, F.: Reduction of the singularities of codimension one singular foliations in dimension three. Ann. Math. (2) 160(3), 907–1011 (2004)
Cutkosky, S.: Monomialization of Morphisms from 3-Folds to Surfaces. Lecture Notes in Mathematics, vol. 1786. Springer, Berlin (2002)
Cutkosky, S.: Local monomialization of analytic maps, preprint. arXiv:1504.01299 [math.AG] (2015)
Hörmander, L.: An Introduction to Complex Analysis in Several Variables. North-Holland Publishing Co., Amsterdam (1973)
Kollàr, J.: Lectures on Resolution of Singularities. Annals of Mathematics Studies, vol. 166. Princeton University Press, Princeton (2007)
McQuillan, M.: Canonical models of foliations. Pure Appl. Math. Q. 4(3), 877–1012 (2008)
McQuillan, M., Panazzolo, D.: Almost Étale resolution of foliations. J. Differ. Geom. 95(2), 279–319 (2013)
Panazzolo, D.: Resolution of singularities of real-analytic vector fields in dimension three. Acta Math. 197(2), 167–289 (2006)
Stefan, P.: Accessibility and foliations with singularities. Bull. Am. Math. Soc. 80, 1142–1145 (1974)
Sussmann, H.: Orbits of families of vector fields and integrability of distributions. Trans. Am. Math. Soc. 180, 171–188 (1973)
Villamayor, O.: Constructiveness of Hironaka’s resolution. Ann. Sci. École Norm. Sup. (4) 22(1), 1–32 (1989)
Villamayor, O.: Resolution in families. Math. Ann. 309(1), 1–19 (1997)
Acknowledgments
I would like to thank Edward Bierstone for the useful suggestions and for reviewing the manuscript. The structure of this manuscript is strongly influenced by him. I would also like to express my gratitude to Daniel Panazzolo for the useful discussions concerning the problem and its applications. Finally, I would like to thank the anonymous reviewer for several very useful comments and, in particular, for suggesting a different title for the manuscript.
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Belotto da Silva, A. Local resolution of ideals subordinated to a foliation. RACSAM 110, 841–862 (2016). https://doi.org/10.1007/s13398-015-0264-0
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DOI: https://doi.org/10.1007/s13398-015-0264-0