2025 Theses Doctoral

On the Hodge Structures of Global Smoothings of Normal Crossing Varieties

Chen, Kuan-Wen

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 Kähler. 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-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 𝗣¹ with disjoint critical locus.