Transversally Pseudoconvex Foliations

Tomassini, Giuseppe, and Venturini, Sergio. "Transversally Pseudoconvex Foliations." Bollettino dell'Unione Matematica Italiana 3.2 (2010): 267-279. <http://eudml.org/doc/290663>.

@article{Tomassini2010,
abstract = {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.},
author = {Tomassini, Giuseppe, Venturini, Sergio},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {267-279},
publisher = {Unione Matematica Italiana},
title = {Transversally Pseudoconvex Foliations},
url = {http://eudml.org/doc/290663},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Tomassini, Giuseppe
AU - Venturini, Sergio
TI - Transversally Pseudoconvex Foliations
JO - Bollettino dell'Unione Matematica Italiana
DA - 2010/6//
PB - Unione Matematica Italiana
VL - 3
IS - 2
SP - 267
EP - 279
AB - 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.
LA - eng
UR - http://eudml.org/doc/290663
ER -