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

TL;DR

This paper proves the non-vanishing conjecture for irregular lc pairs using Chen-Jiang decomposition and extends a key decomposition to klt pairs, aiding in establishing sections of certain line bundles.

Contribution

It applies Chen-Jiang decomposition to verify the non-vanishing conjecture for irregular lc pairs and generalizes the Catanese-Fujita-Kawamata decomposition to klt pairs.

Findings

Non-vanishing conjecture holds for irregular lc pairs under certain conditions.

Extension of the Catanese-Fujita-Kawamata decomposition to klt pairs.

Existence of sections of line bundles in specific cases.

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.