This is a survey chapter on our works related to the rigidity of pseudoconvex domains fibered by open Riemann surfaces according to directional moduli. For a marked open Riemann surface R of finite genus g and a real g-vector \(\mathbf {a}=(a_1, \ldots , a_g)\ne \mathbf {0}\) , we introduce the \(\mathbf {a}\) -span \(\rho _{\mathbf {a}}\) , and establish a new relation between \(\rho _{\mathbf {a}}\) and the set of period matrices of all closings of R. From the viewpoint of several complex variables, a variational formula of \(\rho _{\mathbf {a}}(t)\) is obtained for a smooth family \(\mathcal R\) of open Riemann surfaces \(R(t)\) with a complex parameter t in a disk \(\Delta \) . As an application, we state the subharmonicity of the diameter of the \(\mathbf {a}\) -directional moduli disk for higher genera when \(\mathcal R\) is a two-dimensional pseudoconvex domain fibered by open Riemann surfaces of the same topological type.