On the complex torus, all Chern numbers vanish, but the same is true on the compact complex manifold $G/\Gamma$, given by quotienting a complex Lie group by a cocompact lattice. Such lattices exist in all complex semisimple Lie groups, I believe by results of Mostow. See E. Ghys, Deformation des structures complexes sur les espaces homogènes de $SL_2\mathbb{C}$, J. Riene Angew. Math., 468 (1995), p. 113-138. These complex manifolds admit a holomorphic connection on the tangent bundle.

On the complex torus, all Chern numbers vanish, but the same is true on the compact complex manifold $G/\Gamma$, given by quotienting a complex Lie group by a cocompact lattice. Such lattices exist in all complex semisimple Lie groups, I believe by results of Mostow. See E. Ghys, Deformation des structures complexes sur les espaces homogènes de $SL_2\mathbb{C}$, J. Riene Angew. Math., 468 (1995), p. 113-138. These complex manifolds admit a holomorphic connection on the tangent bundle. They are not Kähler except when $G$ is a complex torus and $\Gamma=\{1\}$.