On complex compact parallelizable manifolds admitting affine structures (Q1032850)

Complex compact holomorphic manifolds of complex dimension \( n \) which are parallelisable are fairly restricted: By a result of \textit{H.C. Wang} [Proc. Am. Math. Soc. 5, 771--776 (1954; Zbl 0056.15403)] they are quotients \( G/D \) of a simply connected complex Lie group \( G \) by a discrete subgroup \(D\). The author gives a classification of such spaces which admit a flat and torsion free affine connection in terms of an injective Lie algebra homomorphism from the Lie algebra of \(G\) into the Lie algebra of the affine group of \(\mathbb{C}^{n}\). He then exhibits complex compact manifolds admitting a holomorphic affine connection but no torsion free holomorphic flat affine connections. His examples are based on the work of \textit{E. Ghys} [J. Reine Angew. Math. 468, 113--138 (1995; Zbl 0868.32023)].