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$ 1$-complete semiholomorphic foliations

Authors: Samuele Mongodi and Giuseppe Tomassini
Journal: Trans. Amer. Math. Soc. 368 (2016), 6271-6292
MSC (2010): Primary 32C15, 32V10, 57R30
Published electronically: December 9, 2015
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Abstract: A semiholomorphic foliation of type $ (n,d)$ is a differentiable real manifold $ X$ of dimension $ 2n+d$, foliated by complex leaves of complex dimension $ n$. In the present work, we introduce an appropriate notion of pseudoconvexity (and consequently, $ q$-completeness) for such spaces, given by the interplay of the usual pseudoconvexity along the leaves, and the positivity of the transversal bundle. For $ 1$-complete real analytic semiholomorphic foliations, we obtain a vanishing theorem for the CR cohomology, which we use to show an extension result for CR functions on Levi flat hypersurfaces and an embedding theorem in $ \mathbb{C}^N$. In the compact case, we introduce a notion of weak positivity for the transversal bundle, which allows us to construct a real analytic embedding in $ \mathbb{CP}^N$.

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Additional Information

Samuele Mongodi
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, Italy

Giuseppe Tomassini
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri, 7, I-56126 Pisa, Italy

Keywords: Semiholomorphic foliations, CR geometry, complex spaces
Received by editor(s): April 14, 2014
Received by editor(s) in revised form: August 12, 2014
Published electronically: December 9, 2015
Additional Notes: The first author was supported by the ERC grant HEVO - Holomorphic Evolution Equations n. 277691.
Article copyright: © Copyright 2015 American Mathematical Society