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Distributions, First Integrals and Legendrian Foliations


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

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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)

    MATH  Google Scholar 

  • 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)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

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Luza, M.F., Rosas, R. Distributions, First Integrals and Legendrian Foliations. Bull Braz Math Soc, New Series 53, 1157–1229 (2022).

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