Transversally Pseudoconvex Foliations

Tomassini, Giuseppe ; Venturini, Sergio

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

Résumé

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.