On the Hodge Structures of Global Smoothings of Normal Crossing Varieties

Let f:X → Δ be a one-parameter semistable degeneration of m-dimensional compact complex manifolds. Assume that each component of the central fiber X₀ is K\"ahler. Then, we provide a criterion for a general fiber to satisfy the ∂∂̄-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-K\"ahler Calabi-Yau threefolds, and Sano's non-K\"ahler 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 P¹ with disjoint critical locus.