The $${\overline{\partial}}$$ operator along the leaves and Guichard’s theorem for a complex simple foliation

Abstract

In (Ann Sc ENS Ser 3 4:361–380, 1887) 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 τ is the translation by 1 on \({{\mathbb C}}\). In this note we prove an analogous 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 γ 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.