Geometry of logarithmic forms and deformations of complex structures

We present a new method to solve certain dbar-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. 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. Our method also plays an important role in Cao–Paun's recent works on the extension of pluricanonical sections and proof of Fujino's injectivity conjecture.
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. Its projective case was originally obtained by Katzarkov–Kontsevich–Pantev in 2008.
This talk is based on a joint work with Kefeng Liu and Xueyuan Wan.