  We present a new method to solve certain $\bar{\partial}$-equations for
logarithmic differential forms by using harmonic integral theory for currents
on Kahler manifolds. The result can be considered as a $\bar{\partial}$-lemma
for logarithmic forms. As applications, we generalize the result of Deligne
about closedness of logarithmic forms, give geometric and simpler proofs of
Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral
sequences at $E_1$-level, as well as certain injectivity theorem on compact
Kahler manifolds.
  Furthermore, for a family of logarithmic deformations of complex structures
on Kahler manifolds, we construct the extension for any logarithmic
$(n,q)$-form on the central fiber and thus deduce the local stability of log
Calabi-Yau structure by extending an iteration method to the logarithmic forms.
Finally we prove the unobstructedness of the deformations of a log Calabi-Yau
pair and a pair on a Calabi-Yau manifold by differential geometric method.
