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Actions of \((\mathbf R,+)\) and \((\mathbf C,+)\) on complex manifolds. (English) Zbl 0872.32021

This article explores some aspects of holomorphic actions of R and C on a complex manifold \(M\). One can view such actions as those associated to complete holomorphic vector fields or “flows.” If \(M\) is Stein, the author proves statements about the fundamental domain and the existence of generic orbit types for a holomorphic R action, and in special cases, about the extension of R actions to C actions. The article includes sections on symplectic actions on \({\mathbf C}^{2n}\), obstructions to completeness of certain Hamiltonian vector fields on \({\mathbf C}^2\), and special results about automorphisms and 1-dimensional actions on \({\mathbf C}^n\) and complements \({\mathbf C}^n\setminus A\) where \(A\) is a “tame” analytic subvariety.

MSC:

32M05 Complex Lie groups, group actions on complex spaces
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