# Some non-vanishing results on log canonical pairs of dimension 4

## TL;DR

This paper establishes non-vanishing and semi-ampleness criteria for log canonical pairs of dimension 4, assuming LMMP conjectures, with implications for uniruled varieties and irregular cases.

## Contribution

It provides new non-vanishing and semi-ampleness results for log canonical pairs in dimension 4 under certain conjectural assumptions.

## Key findings

- If $X$ is a uniruled 4-fold, then $ ext{kod}(K_X+ riangle) extgreater{}0$.
- For 4-folds with positive irregularity, $K_X+ riangle$ is semi-ample.
- Results depend on LMMP conjectures in lower dimensions.

## Abstract

Let $(X,\Delta)$ be a log canonical pair over $\mathbb{C}$ with $X$ a normal projective variety, $\Delta$ an effective $\mathbb{Q}$-divisor, and $K_X+\Delta$ nef. We give a non-vanishing criterion for $K_X+\Delta$ in dimension $n$ with $X$ uniruled, assuming various conjectures of LMMP in dimensions (up to) $n-1$ or $n$, and a semi-ampleness criterion in the irregular case. In particular, we obtain that if $X$ is a uniruled $4$-fold, then $\kappa(K_X+\Delta)\geq 0$ and if $X$ is a $4$-fold with $q(X)>0$, then $K_X+\Delta$ is semi-ample.