On the Hodge Structures of Global Smoothings of Normal Crossing Varieties

Let $f:X\to \mathrm{\Delta }$ be a one-parameter semistable degeneration of $m$-dimensional compact complex manifolds. Assume that each component of the central fiber $X_0$ is Kähler. Then, we provide a criterion for a general fiber to satisfy the $\mathrm{\partial }\bar{\mathrm{\partial }}$-lemma and a formula to compute the Hodge index on the middle cohomology of the general fiber in terms of the topological conditions/invariants on the central fiber. We apply our theorem to several examples, including the global smoothing of $m$-fold ODPs, Hashimoto-Sano’s non-Kähler Calabi-Yau threefolds, and Sano’s non-Kähler Calabi-Yau $m$-folds. To deal with the last example, we also prove a Lefschetz-type theorem for the cohomology of the fiber product of two Lefschetz fibrations over ${\mathbb{P}}^1$ with disjoint critical locus.