The introduction: “We give a survey of our recent results on complex-parallelizable manifolds, most of which are contained in [J. Winkelmann, ‘Complex analytic geometry of complex parallelizable manifolds’, Habil. schrift (1994)]. For our purposes it is useful to call a complex manifold parallelizable if it is biholomorphic to a quotient of a complex Lie group \(G\) by a discrete subgroup \(\Gamma\). By a result of H.-C. Wang [Proc. Am. Math. Soc. 5, 771-776 (1954; Zbl 0056.15403)] for a compact complex manifold this condition is equivalent to the assumption that the tangent bundle is holomorphically trivial. A compact complex parallelizable manifold is Kähler if and only if \(G\) is commutative, i.e. if and only if it is a torus”.
For the entire collection see [Zbl 0903.00037].