Transversally Pseudoconvex Foliations

Giuseppe Tomassini; Sergio Venturini

Bollettino dell'Unione Matematica Italiana (2010)

  • Volume: 3, Issue: 2, page 267-279
  • ISSN: 0392-4041

Abstract

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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 { λ j } on the the fibres such that the tangential (1,1)-form Ω = { λ j ¯ λ j - 2 ¯ λ j λ j } is positive. This condition is of a special interest if the foliation X is 1 complete i.e. admits a smooth exhaustion function ϕ which is strongly plusubharmonic along the leaves. In this situation we prove that there exist an open neighbourhood U of X in the complexification X ~ of X and a non negative smooth function u : U 𝐑 which is plurisubharmonic in U , strongly plurisubharmonic on U X and such that X is the zero set of u . This result has many implications: every compact sublevel X ¯ c = { x X : ϕ c } is a Stein compact and if S ( X ) is the algebra of smooth CR functions on X , the restriction map S ( X ) 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 𝐂 2 n + 3 by a CR map and the sheaf S = S X of germs of smooth CR functions on X is cohomologically trivial.

How to cite

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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 -

References

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  1. ANDREOTTI, A. - GRAUERT, H., Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, 90 (1962), 193-259. Zbl0106.05501MR150342
  2. FREEMAN, M., Tangential Cauchy-Riemann equations and uniform approximation, Pacific J. Math., 33 (1970), 101-108. Zbl0184.31103MR264117
  3. GIGANTE, G. - TOMASSINI, G., Foliations with complex leaves, Diff. Geom. Appl., 5 (1995), 33-49. Zbl0843.32012MR1319934DOI10.1016/0926-2245(95)00004-N
  4. HÖRMANDER, L., An introduction to complex analysis in several variables, D. Van Nostrand, Princeton (New Yersey, 1965). MR203075
  5. KOHN, J. J., Global regulatity for ¯ on weakly pseudo convex manifolds, Trans. Am. Math. Soc., 181 (1962), 193-259. MR344703DOI10.2307/1996633

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