On any compact complex manifold, the set of all global holomorphic vector fields is a finite dimensional Lie algebra. They span the tangent space at every point just when each component of the manifold is a homogeneous for the action of its biholomorphism group. Read the book by Dmitri Akhiezer, **Lie  group  actions  in  complex  analysis**, Aspects of Mathematics, vol.  E27,  Friedr.  Vieweg,  Braunschweig and Wiesbaden,  1995,  vii +
201 pp., $49.00, ISBN 3-528-06420-X.

If $c_1 \le 0$, then they are tori, by Bochner, I believe. Check Wang's paper on homogeneous complex manifolds.

They are strongly dominable, since the Lie group of biholomorphisms has an exponential map.