Quasi-isometry and deformations of Calabi-Yau manifolds

We prove several formulas related to Hodge theory and the Kodaira-Spencer-Kuranishi deformation theory of K"ahler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on Calabi-Yau manifolds and also a construction of global canonical family of holomorphic $(n,0)$-forms on the deformation spaces of Calabi-Yau manifolds. Similar constructions are also applied to the deformation spaces of compact K"ahler manifolds.