Compact Kähler manifolds satisfy several nice cohomological properties
such as Hodge symmetry and Hodge-Riemann bilinear relations.
Friedman and Li recently showed that non-Kähler Calabi-Yau 3-folds
which are obtained by conifold transitions of projective ones satisfy
such properties.
In this talk, I will present examples of non-Kähler Calabi-Yau
manifolds with such properties by smoothing normal crossing varieties.