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

Abstract

By applying the Chen-Jiang decomposition, we prove that the non-vanishing conjecture holds for an lc pair \((X, \Delta)\), 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 \((X, \Delta)\), which leads to the existence of sections of \(K_X + \Delta\) in certain situations.