Dado una acción holomorfa afín φ:Aff(C) x M →M del grupo Aff(C) del grupo sobre una variedad compleja M. Se sabe que esta acción asocia dos campos holomorfos completos X e Y con X periódica de periodo 2πi y que están relacionados por el corchete de Lie mediante la relación. [X,Y]= – Y. El conjunto singular de φ es sing(Zφ) está dado por los puntos. Se resuelve el siguiente problema: En sing (Zφ) las órbitas de φ son biholomorfas a un punto, a C, a C^* o al toro complejo T y en M – sing(Zφ) y en las órbitas son biholomorfas a .

In this article, several generalizations in affine spaces are studied. First, the notion of affine mappings to bilinear mappings defined in affine spaces is explored, referred to as affine bilinear mappings. Subsequently, differentiable actions of a Lie group on affine spaces are defined, and their isotropy group, orbit space, and set of fixed points are examined. Finally, the concept of tensor product between vector spaces is extended to the tensor product between affine spaces.

In this work, we study actions of the Lie group SL(2,C) on a complex manifold of dimension three or higher. It is demonstrated that these types of actions induce three complete holomorphic vector fields, one of which is periodic, and that there exists a particular relationship between them, given by the Lie bracket, which generates a singular holomorphic foliation of codimension two. Subsequently, the types of singularities are classified, and the normal forms of these vector fields are obtained in a neighborhood of each singular point of the foliation.