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

Let X be a 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 h0(m(KX+L))>0 for any polarized varieties (X,L) with dimX=4 such that X is a normal Gorenstein projective variety with only terminal singularities and KX+L is nef or for any polarized varieties (X,L) with dimX=4 such that X is smooth and κ(KX+L)≥0.