The group of holomorphic diffeomorphisms, Aut(M), of a compact complex manifold M is a Lie group of
finite dimension. To provide bounds for the dimension of these groups is a classical problem in complex
analysis. It is well known that the dimension of Aut(M) cannot be bounded in terms of the dimension of M
solely. However several important problems arise once specific constraints are imposed on the manifold M.
For example, the case of homogeneous manifolds has been intensively studied in connection with which is
sometimes called Remmert conjecture. Another interesting situation corresponds to the case of algebraic
manifolds whose Picard group is Z (the Hwang-Mok problem).

There is an evident relation between bounds for the order of the zeros of holomorphic vector fields on M
and bounds for the dimension of Aut(M). In this sense, results on singularities of holomorphic vector fields
(particularly on specific questions concerning the extent to which a singularity of a vector field can be
degenerate) have implications to problem mentioned above. We will discuss some recent results in this
direction.