On Hodge structures of compact complex manifolds with semistable degenerations

TL;DR

This paper explores when key Hodge-theoretic properties like symmetry and the Hodge-Riemann relations hold for compact complex manifolds with semistable degenerations, especially in specific smoothing cases.

Contribution

It establishes conditions under which Hodge symmetry and Hodge-Riemann relations are valid for manifolds arising from smoothings of SNC varieties without triple intersections.

Findings

Hodge symmetry holds when monodromy induces isomorphisms on graded pieces.

Hodge-Riemann relations are valid on H^3 of certain compact complex 3-folds.

Results apply to manifolds obtained as smoothings of specific SNC varieties.

Abstract

Compact K\"{a}hler 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 pieces of the weight filtrations of the limiting mixed Hodge structures. We also show the Hodge-Riemann relations on H^3 of compact complex 3-folds with such semistable degenerations under some conditions.