On Hodge structures of compact complex manifolds with semistable degenerations

Compact K\"ahler manifolds satisfy several nice Hodge-theoretic properties such as the Hodge symmetry, the Hard Lefschetz property and the Hodge--Riemann bilinear relations, etc. In this note, we investigate when such nice properties hold on compact complex manifolds with semistable degenerations. For compact complex manifolds which can be obtained as smoothings of SNC varieties without triple intersection locus, we show the Hodge symmetry when the monodromy logarithm induces isomorphisms on the associated graded. We also show the Hodge--Riemann relations on H³ of non-K\"ahler Calabi--Yau 3-folds with such semistable degenerations.