# Distributions, First Integrals and Legendrian Foliations

## 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

\begin{aligned} dz=(y_1dx_1-x_1dy_1)+\dots +(y_mdx_m-x_mdy_m), \end{aligned}

which defines the canonical contact structure on $${\mathbb {C}}{\mathbb {P}}^{2m+1}$$. Another example is the Darboux distribution

\begin{aligned} dz=x_1dy_1+\dots +x_mdy_m, \end{aligned}

which gives the normal local form of any contact structure. Given a germ $${\mathcal {D}}$$ of holomorphic distribution with separated variables in $$({\mathbb {C}}^n,0)$$, we show that there exists , for some $$\kappa \in {\mathbb {Z}}_{\ge 0}$$ related to the Taylor coefficients of $${\mathcal {D}}$$, a holomorphic submersion

\begin{aligned} H_{{\mathcal {D}}}:({\mathbb {C}}^n,0)\rightarrow ({\mathbb {C}}^{\kappa },0) \end{aligned}

such that $${\mathcal {D}}$$ is completely non-integrable on each level of $$H_{{\mathcal {D}}}$$. Furthermore, we show that there exists a holomorphic vector field Z tangent to $${\mathcal {D}}$$, such that each level of $$H_{{\mathcal {D}}}$$ 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 $${\mathcal {D}}$$ are the same. Between several other results, we show that the canonical contact structure on $${\mathbb {C}}{\mathbb {P}}^{2m+1}$$ supports a Legendrian holomorphic foliation whose generic leaves are dense in $${\mathbb {C}}{\mathbb {P}}^{2m+1}$$. So we obtain examples of injectively immersed Legendrian holomorphic open manifolds that are everywhere dense.

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## References

• Alarcón, A., Forstneric, F., López, F.J.: Holomorphic Legendrian curves. Compos. Math. 153, 1945–1986 (2017)

• Alarcón, A., Forstneric, F., Larusson, F.: Holomorphic Legendrian curves in $${\mathbb{C}}{\mathbb{P}}(3)$$ and superminimal surfaces in $${\mathbb{S}}^4$$. arXiv:1910.12996 (2019)

• Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren der mathematischen Wissenschaften, vol. 250. Springer, New York (1983)

• Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60. Springer, New York (1989)

• Bonnet, P.: Families of k-derivations on k-algebras. J. Pure Appl. Algebra 199(1–3), 11–26 (2005)

• Buczynski, J.: Algebraic Legendrian varieties. arXiv:0805.3848v2 (2008)

• Chow, W.L.: Uber Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann. 117, 98–105 (1939)

• Darboux, G.: Sur le problème de Pfaff. Bull. Sci. Math. 6, 14–36, 49–68 (1882)

• Godbillon, C.: Géométrie différentielle et mécanique analytique, p. 183. Hermann, Paris (1969)

• Gromov, M.: Carnot-Carathéodory Spaces Seen from Within, in Sub-Riemannian Geometry. Progr. Math., vol. 144. Birkhauser, Basel, pp. 79–323 (1996)

• Nagata, M., Nowicki, A.: Rings of constants for $$k$$-derivations in $$k[x_1,\dots, x_n]$$. J. Math. Kyoto Univ. 28(1), 111–118 (1998)

• Nowicki, A.: Rings and fields of constants for derivations in characteristic zero. J. Pure Appl. Algebra 96(1), 47–55 (1994)

• Singer, M.F.: Liouvillian first integrals of differential equations. Trans. Am. Math. Soc. 333, 673–688 (1992)

• Zorich, V.A.: Holomorphic distributions and connectivity by integral curves of distributions. arXiv:1907.05610 (2019)

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Correspondence to Maycol Falla Luza.

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Maycol Falla Luz was partially supported by CAPES-COFECUB Ma932/19. Rudy Rosas was supported by the Vicerrectorado the Investigación de la Pontificia Universidad Católica del Perú

Luza, M.F., Rosas, R. Distributions, First Integrals and Legendrian Foliations. Bull Braz Math Soc, New Series 53, 1157–1229 (2022). https://doi.org/10.1007/s00574-022-00300-0