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Distributions, First Integrals and Legendrian Foliations.
- Source :
-
Bulletin of the Brazilian Mathematical Society . Dec2022, Vol. 53 Issue 4, p1157-1229. 73p. - Publication Year :
- 2022
-
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 d z = (y 1 d x 1 - x 1 d y 1) + ⋯ + (y m d x m - x m d y m) , which defines the canonical contact structure on C P 2 m + 1 . Another example is the Darboux distribution d z = x 1 d y 1 + ⋯ + x m d y m , which gives the normal local form of any contact structure. Given a germ D of holomorphic distribution with separated variables in (C n , 0) , we show that there exists , for some κ ∈ Z ≥ 0 related to the Taylor coefficients of D , a holomorphic submersion H D : (C n , 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. Between several other results, we show that the canonical contact structure on C P 2 m + 1 supports a Legendrian holomorphic foliation whose generic leaves are dense in C P 2 m + 1 . So we obtain examples of injectively immersed Legendrian holomorphic open manifolds that are everywhere dense. [ABSTRACT FROM AUTHOR]
- Subjects :
- *INTEGRALS
*FOLIATIONS (Mathematics)
*VECTOR fields
*MEROMORPHIC functions
Subjects
Details
- Language :
- English
- ISSN :
- 16787544
- Volume :
- 53
- Issue :
- 4
- Database :
- Academic Search Index
- Journal :
- Bulletin of the Brazilian Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 159896148
- Full Text :
- https://doi.org/10.1007/s00574-022-00300-0