The compact torus S1 × S1 has a structure of Riemann surfaceand therefore is a complex projective manifold. On product of odd dimensionalspheres S2p+1 × S2q+1 with p > 0 or q > 0, complex structures were obtainedby H. Hopf (1948) and Calabi-Eckmann (1953). These complex manifolds areone of the first examples of non-K¨ahler, and hence non-projective, compactcomplex manifolds.The aim of this talk is to describe a general construction of a class ofnon-K¨ahler compact complex manifolds. Let G be a complex linear algebriacgroup and let K be a maximal compact subgroup of G. Any holomorphicprincipal G-bundle EG over a complex manifold admits a smooth reduction ofthe structure group from G to K. We will show that the total space EK of thesmooth principal K-bundle, corresponding to this reduction, admits a complexstructure. In most cases, the complex manifold EK will be non-K¨ahler. Thistalk is based on a joint work with Mainak Poddar.
School of Mathematical Sciences
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