Equivariant deformations of meromorphic actions on compact complex manifolds

Abstract

Meromorphicity is the most basic property for holomorphic ${\mathbb C}^*$ -actions on compact complex manifolds. We prove that the meromorphicity of ${\mathbb C}^*$ -actions on compact complex manifolds are not necessarily preserved by small deformations, if the complex dimension of complex manifolds is greater than two. In contrast, we also show that the meromorphicity of ${\mathbb C}^*$ -actions on compact complex surface depends only on the topology (the first Betti number) of the surface. We construct such examples of dimension greater than two by studying an equivariant deformation of certain complex threefold, so called a twistor space.