On \(\mathbb{C}\)-actions on compact complex manifolds with many compact orbits (Q1320366)

The author proves the following theorem. Suppose \(X\) is a compact complex manifold which admits a holomorphic action of the Lie group \(G = (\mathbb{C},+)\) such that there is a \(G\)-invariant set \(\Omega \subset X\) of positive Lebesgue measure in which every \(G\)-orbit is compact. Then the \(G\)-action on \(X\) fibers through a torus action. This result was proved in the case \(\Omega = X\) by \textit{H. Holmann} [Comment. Math. Helv. 52, 251-257 (1977; Zbl 0353.32034)], and in the case \(X\) is Kähler a similar statement follows from the results of \textit{D. Snow} [Arch. Math. 37, 364-371 (1981; Zbl 0468.57030)]. The author also gives examples which show that the corresponding statements are false if one has a differentiable \(\mathbb{R}\)-action or if \(X\) is not compact.