Finite distance problem on the moduli of non-Kähler Calabi--Yau $\partial\bar{\partial}$-threefolds

Abstract: In this article, we study the finite distance problem with respect to the period-map metric on the moduli of non-K\"{a}hler Calabi--Yau $\partial\bar{\partial}$-threefolds via Hodge theory. We extended C.-L. Wang's finite distance criterion for one-parameter degenerations to the present setting. As a byproduct, we also obtained a sufficient condition for a non-K\"{a}hler Calabi--Yau to support the $\partial\bar{\partial}$-lemma which generalizes the results by Friedman and Li. We also proved that the non-K\"{a}hler Calabi--Yau threefolds constructed by Hashimoto and Sano support the $\partial\bar{\partial}$-lemma.