TAILIEUCHUNG - Đề tài " Reduction of the singularities of codimension one singular foliations in dimension three "

Reduction of the singularities of codimension one singular foliations in dimension three By Felipe Cano Contents 0. Introduction 1. Blowing-up singular foliations . Adapted singular foliations . Permissible centers . Vertical invariants . First properties of presimple singularities 2. Global strategy . Reduction to presimple singularities. Statement . Good points. Bad points. Equi-reduction . Finiteness of bad points . The influency locus . The local control theorem . Destroying cycles . Global criteria of blowing-up 3. Local control . Two-dimensional differential idealistic exponents . | Annals of Mathematics Reduction of the singularities of codimension one singular foliations in dimension three By Felipe Cano Annals of Mathematics 160 2004 907 1011 Reduction of the singularities of codimension one singular foliations in dimension three By Felipe Cano Contents 0. Introduction 1. Blowing-up singular foliations . Adapted singular foliations . Permissible centers . Vertical invariants . First properties of presimple singularities 2. Global strategy . Reduction to presimple singularities. Statement . Good points. Bad points. Equi-reduction . Finiteness of bad points . The influency locus . The local control theorem . Destroying cycles . Global criteria of blowing-up 3. Local control . Two-dimensional differential idealistic exponents . Maximal contact . Local control in the first m-stable cases . Adapted multiplicity bigger than the adapter order . The resonant m-stable case . Adapted multiplicity equal to the adapted order . Local control and characteristic polygons . Reduction to nondicritical-like behaviour . Normalized coordinate data . Characteristic polygons . Break indices . The jumping situation 4. Getting simple singularities . Jordanization . The singular locus . Elimination of the Jordan blocks . Killing the resonances . The final normal crossings Appendix About simple singularities 908 FELIPE CANO 0. Introduction The reduction of the singularities of a codimension one holomorphic foliation over an ambient space of dimension two has been achieved by Seidenberg in 26 . Here we give a complete answer to this problem over an ambient space of dimension three as stated in the following theorem. Theorem reduction to simple singularities . Let X be a three -dimensional germ around a compact analytic subset of nonsingular complex analytic space. Let F be a holomorphic singular foliation of codimension one and D a normal crossings divisor on X. Then .

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