A KAM phenomenon for singular holomorphic vector fields
Publications Mathématiques de l'IHÉS, Tome 102 (2005), p. 99-165
Let X be a germ of holomorphic vector field at the origin of 𝐂 n and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form “resonant monomials = constants”. Such a biholomorphism conjugates the restriction of X to one of its invariant varieties to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of “frequencies” defining the invariant sets is of positive measure.
@article{PMIHES_2005__102__99_0,
     author = {Stolovitch, Laurent},
     title = {A KAM phenomenon for singular holomorphic vector fields},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Springer},
     volume = {102},
     year = {2005},
     pages = {99-165},
     doi = {10.1007/s10240-005-0035-0},
     zbl = {1114.37026},
     mrnumber = {2217052},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2005__102__99_0}
}
Stolovitch, Laurent. A KAM phenomenon for singular holomorphic vector fields. Publications Mathématiques de l'IHÉS, Tome 102 (2005) pp. 99-165. doi : 10.1007/s10240-005-0035-0. http://www.numdam.org/item/PMIHES_2005__102__99_0/

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