Compact Kaehler and projective manifolds with holomorphic tangent vector fields

The birational classification of projective manifolds with holomorphic tangent vector fields was developed by F. Severi, R. Hall and D. Lieberman.
Adapting Bogomolov's technique for the study of varieties with trivial canonical bundle, one may use the Albanese mapping to obtain the biholomorphic classification of non-uniruled projective manifolds with holomorphic tangent vector fields.
These results may be extended to compact Kaehler manifolds, using small deformations of the complex structure. They show that the study of the dynamics of holomorphic vector fields in them reduces to the case of rational varieties.