Two Applications of the \(\partial \overline \partial \)-Hodge Theory

Using Hodge theory and Banach fixed point theorem, Liu and Zhu developed a global method to deal with various problems in deformation theory. In this note, the authors generalize Liu-Zhu’s method to treat two deformation problems for non-Kähler manifolds. They apply the \(\partial \overline \partial \) -Hodge theory to construct a deformation formula for (p, q)-forms of compact complex manifold under deformations, which can be used to study the Hodge number of complex manifold under deformations. In the second part of this note, by using the \(\partial \overline \partial \) -Hodge theory, they provide a simple proof of the unobstructed deformation theorem for the non-Kähler Calabi-Yau \(\partial \overline \partial \) -manifolds.