Remarks on the Direct Image Sheaf of Logarithmic Pluricanonical Bundles and the Non-Vanishing Conjecture

By applying the Chen-Jiang decomposition, we prove that the non-vanishing conjecture holds for an lc pair $(X,\mathrm{\Delta })$, where $X$ is an irregular variety, provided it holds for lower-dimensional varieties. In the second part, we extend the Catanese-Fujita-Kawamata decomposition to the klt case $(X,\mathrm{\Delta })$, which leads to the existence of sections of $K_X+\mathrm{\Delta }$ in certain situations.