{"title":"Deforming the weighted-homogeneous foliation, and trivializing families of semi-weighted homogeneous ICIS","authors":"Dmitry Kerner, Rodrigo Mendes","doi":"arxiv-2409.09764","DOIUrl":null,"url":null,"abstract":"Let X_o be a weighted-homogeneous complete intersection germ in (R^N,o) or\n(C^N,o), with arbitrary singularities, possibly non-reduced. Take the foliation\nof the ambient space by weighted-homogeneous real arcs, \\ga_s. Take a deformation of X_o by higher order terms, X_t. Does the foliation\n\\ga_s deform compatibly with X_t? We identify the ``obstruction locus\", \\Sigma\nin X_o, outside of which such a deformation does exist, and possesses\nexceptionally nice properties. Using this deformed foliation we construct a contact trivialization of the\nfamily of defining equations by a homeomorphism that is real analytic (resp.\nNash) off the origin, differentiable at the origin, whose presentation in\nweighted-polar coordinates is globally real-analytic (resp. globally Nash), and\nwith controlled Lipschitz/C^1-properties.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09764","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let X_o be a weighted-homogeneous complete intersection germ in (R^N,o) or
(C^N,o), with arbitrary singularities, possibly non-reduced. Take the foliation
of the ambient space by weighted-homogeneous real arcs, \ga_s. Take a deformation of X_o by higher order terms, X_t. Does the foliation
\ga_s deform compatibly with X_t? We identify the ``obstruction locus", \Sigma
in X_o, outside of which such a deformation does exist, and possesses
exceptionally nice properties. Using this deformed foliation we construct a contact trivialization of the
family of defining equations by a homeomorphism that is real analytic (resp.
Nash) off the origin, differentiable at the origin, whose presentation in
weighted-polar coordinates is globally real-analytic (resp. globally Nash), and
with controlled Lipschitz/C^1-properties.