Solving equations with Hodge theory

We treat two quite different problems related to changes of complex structures on K\"ahler manifolds by using global geometric method. First, by using operators from Hodge theory on compact K\"ahler manifold, we present a closed explicit extension formula for holomorphic canonical forms in different complex structures. As applications, we give a closed explicit formula for certain canonical sections of Hodge bundles on marked and polarized moduli spaces of projective manifolds, and provide a closed explicit extension formula for holomorphic pluricanonical forms under certain natural conditions. Second, by using the operators in $L^2$-Hodge theory on Poincar\'e disk, we present a simple and unified method to solve the Beltrami equations with measurable coefficients for quasi-conformal maps.