  We study the Fujita-type conjecture proposed by Popa and Schnell. We obtain
an effective bound on the global generation of direct images of pluri-adjoint
line bundles on the regular locus. We also obtain an effective bound on the
generic global generation for a Kawamata log canonical $\mathbb{Q}$-pair. We
use analytic methods such as $L^2$ estimates, $L^2$ extensions and injective
theorems of cohomology groups.
