On the Hodge Structures of Global Smoothings of Normal Crossing Varieties

TL;DR

This paper establishes criteria for when the general fiber of a semistable degeneration of complex manifolds satisfies the $ar{ ext{d}} ext{d}$-lemma and provides formulas for Hodge indices, with applications to non-Kähler Calabi-Yau varieties.

Contribution

It offers new topological criteria and formulas for Hodge structures in degenerations, extending understanding to non-Kähler Calabi-Yau varieties and fiber products.

Findings

Criteria for the $ar{ ext{d}} ext{d}$-lemma in general fibers.

Formula for Hodge index in terms of central fiber invariants.

Application to non-Kähler Calabi-Yau examples.

Abstract

Let $f:X \rightarrow \Delta $ be a one-parameter semistable degeneration of $m$-dimensional compact complex manifolds. Assume that each component of the central fiber $X_0$ is K\"ahler. Then, we provide a criterion for a general fiber to satisfy the $\partial\overline{\partial}$-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-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 $\mathbb{P}^1$ with disjoint critical locus.