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The \({\overline{\partial}}\) operator along the leaves and Guichard’s theorem for a complex simple foliation. (English) Zbl 1209.32021

Author’s abstract: In [Ann. de l’Éc. Norm. (3) IV. 361–380 (1887; JFM 19.0344.01)] C. Guichard proved that, for any holomorphic function \(g\) on \({{\mathbb C}}\), there exists a holomorphic function \(h\) (on \({{\mathbb C}}\)) such that \({h - h \circ \tau = g}\) where \(\tau \) is the translation by 1 on \({{\mathbb C}}\). In this note, we prove an analog of this theorem in a more general situation. Precisely, let \({(M,{\mathcal F})}\) be a complex simple foliation whose leaves are simply connected non compact Riemann surfaces and \(\gamma \) an automorphism of \({{\mathcal F}}\) which fixes each leaf and acts on it freely and properly. Then, the vector space \({{\mathcal H}_{\mathcal F}(M)}\) of leafwise holomorphic functions is not reduced to functions constant on the leaves, and, for any \({g \in {\mathcal H}_{\mathcal F}(M)}\), there exists \({h \in {\mathcal H}_{\mathcal F}(M)}\) such that \({h - h \circ \gamma = g}\). From the proof of this theorem, we derive a foliated version of Mittag-Leffler Theorem.

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32G05 Deformations of complex structures
58A30 Vector distributions (subbundles of the tangent bundles)

Citations:

JFM 19.0344.01
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References:

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