By a Calabi-Yau manifold, I mean a compact complex manifold whose canonical bundle is trivial and $$H^i (X, O_X) = H^0(X, \Omega_X^i) = 0$$ for $0 < i < \dim X$. Infinitely many topological types of non-Kähler Calabi-Yau 3-folds have been constructed. Are there any examples of non-Kähler Calabi-Yau 4-folds or higher dimension?
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$\begingroup$ There are torus $T^2$ fibrations over a Kahler CY base constructed in a paper of Goldstein-Prokushkin. And maybe also compact complex Lie groups (having trivial canonical bundle but I don't think have a Kahler metric). $\endgroup$– Chris GerigCommented Oct 5, 2019 at 15:40
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$\begingroup$ @ChrisGerig >>compact complex Lie groups<< they are indeed not even symplectic unless abelian, but admit a plenty of holomorphic forms $\endgroup$– Dmitrii KorshunovCommented Oct 5, 2019 at 22:17
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