The (partial derivative)over-bar operator along the leaves and Guichard's theorem for a complex simple foliation

作者：Alaoui Aziz El Kacimi^*

来源：Mathematische Annalen, 2010, 347(4): 885-897.

DOI：10.1007/s00208-009-0459-9

In (Ann Sc ENS Ser 3 4:361-380, 1887) Guichard proved that, for any holomorphic function g on C, there exists a holomorphic function h (on C) such that h - h circle tau = g where tau is the translation by 1 on C. In this note we prove an analogous of this theorem in a more general situation. Precisely, let (M, F) be a complex simple foliation whose leaves are simply connected non compact Riemann surfaces and. an automorphism of F which fixes each leaf and acts on it freely and properly. Then, the vector space H(F) (M) of leafwise holomorphic functions is not reduced to functions constant on the leaves and for any g is an element of H(F) (M), there exists h is an element of H(F) (M) such that h - h circle gamma = g. From the proof of this theorem we derive a foliated version of Mittag-Leffler Theorem.

• 出版日期2010-8