Effective non-vanishing of global sections of multiple adjoint bundles for polarized 4-folds

Let X be a smooth complex projective variety of dimension 4 and let L be an ample line bundle on X. In this paper, we study a natural number m such that h^{0}(m(K_{X}+L))>0 for any polarized 4-folds (X,L) with \kappa(K_{X}+L)\geq 0.