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

In this article, we study the finite distance problem with respect to the period-map metric on the moduli of non-Kähler Calabi–Yau $\mathrm{\partial }\bar{\mathrm{\partial }}$-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-Kähler Calabi–Yau to support the $\mathrm{\partial }\bar{\mathrm{\partial }}$-lemma which generalizes the results by Friedman and Li. We also proved that the non-Kähler Calabi–Yau threefolds constructed by Hashimoto and Sano support the $\mathrm{\partial }\bar{\mathrm{\partial }}$-lemma.