Paper detail

The $\partial\bar{\partial}$-lemma for general Clemens manifolds

We show that the $\partial\bar{\partial}$-lemma holds for the non-Kähler compact complex manifolds of dimension three with trivial canonical bundle constructed by Clemens as deformations of Calabi-Yau threefolds contracted along smooth rational curves with normal bundle of type $(-1, -1)$, at least on an open dense set in moduli. The proof uses the mixed Hodge structure on the singular fibers and an analysis of the variation of the Hodge filtration for the smooth fibers.