Geodesic
Transversally Pseudoconvex Foliations
Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 2, pp. 267-279.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We consider real analytic foliations $X$ with complex leaves of transversal dimension one and we give the notion of transversal pseudoconvexity. This amounts to require that the transverse bundle $N_{F}$ to the leaves carries a metric $\{\lambda_{j}\}$ on the the fibres such that the tangential (1,1)-form $\Omega = \{\lambda_{j} \bar{\partial}\partial\lambda_{j} - 2\bar{\partial}\lambda_{j}\partial\lambda_{j}\}$ is positive. This condition is of a special interest if the foliation $X$ is 1 complete i.e. admits a smooth exhaustion function $\phi$ which is strongly plusubharmonic along the leaves. In this situation we prove that there exist an open neighbourhood $U$ of $X$ in the complexification $\widetilde{X}$ of $X$ and a non negative smooth function $u : U \to \mathbf{R}$ which is plurisubharmonic in $U$, strongly plurisubharmonic on $U \setminus X$ and such that $X$ is the zero set of $u$. This result has many implications: every compact sublevel $\overline X_{c} = \{ x \in X : \phi \le c \}$ is a Stein compact and if $S(X)$ is the algebra of smooth CR functions on $X$, the restriction map $S(X) \to S(X_{c})$ has a dense image (Theorem 4.1); a transversally pseudoconvex, 1-complete, real analytic foliation $X$ with complex leaves of dimension $n$ properly embeds in $\mathbf{C}^{2n+3}$ by a CR map and the sheaf $S = S_{X}$ of germs of smooth CR functions on $X$ is cohomologically trivial. 
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     author = {Tomassini, Giuseppe and Venturini, Sergio},
     title = {Transversally {Pseudoconvex} {Foliations}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {267--279},
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     volume = {Ser. 9, 3},
     number = {2},
     year = {2010},
     zbl = {1197.32014},
     mrnumber = {2666358},
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     url = {http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_2_a1/}
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Tomassini, Giuseppe; Venturini, Sergio. Transversally Pseudoconvex Foliations. Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 2, pp. 267-279. http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_2_a1/

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