Group actions and non-K\"ahler complex manifolds
2016
New constructions of non-K\"ahler complex manifolds are presented. Let $H$ be a complex linear algebraic group and let $K$ be a maximal compact Lie subgroup of $H$. Let $\mathcal{E}$ be a smooth principal $K$-bundle $E_K \rightarrow M$ over a complex manifold $M$. If $\mathcal{E}$ can be obtained by a smooth reduction of structure group from a holomorphic principal $H$-bundle over $M$, then $E_K$ (respectively, $E_K \times S^1$) admits an integrable complex structure if $K$ has even dimension (respectively, odd dimension). As a consequence, the total space of the unitary frame bundle associated to any holomorphic vector bundle of even rank admits a complex analytic structure which is not K\"ahler. New complex manifolds are also derived from proper holomorphic actions of complex linear algebraic groups on complex manifolds. In particular, non-K\"ahler complex manifolds associated to effective complex analytic orbifolds are constructed.
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