Distributions, first integrals and Legendrian foliations

2022/05/18 by Luza, Maycol Falla, Rosas, Rudy
#32M25 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2205.08947

Abstract

We study germs of holomorphic distributions with "separated variables'. In codimension one, a well know example of this kind of distribution is given by the canonical contact structure on ℙ^2m+1 . Another example is the Darboux distribution, which gives the normal local form of any contact structure. Given a germ D of holomorphic distribution with separated variables in (ℂⁿ,0), we show that there exists , for some κ∈ ℤ_{≥ 0} related to the Taylor coefficients of D, a holomorphic submersion H_D: (ℂⁿ,0) → (ℂ^κ,0) such that D is completely non-integrable on each level of H_D. Furthermore, we show that there exists a holomorphic vector field Z tangent to D, such that each level of H_D contains a leaf of Z that is somewhere dense in the level. In particular, the field of meromorphic first integrals of Z and that of D are the same.

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