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 Akheizer's book.

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.