Non-vanishing implies numerical dimension one abundance

TL;DR

This paper demonstrates that the non-vanishing conjecture leads to the abundance conjecture under certain conditions and proves the abundance conjecture in dimensions up to five for specific cases.

Contribution

It establishes a link between the non-vanishing and abundance conjectures and proves the latter in low dimensions for certain numerical conditions.

Findings

Non-vanishing implies abundance when ν ≤ 1.

Abundance conjecture proven in dimension ≤ 5 for κ ≥ 0 and ν ≤ 1.

Conditional and unconditional results in algebraic geometry.

Abstract

We show that the non-vanishing conjecture implies the abundance conjecture when $\nu\leq 1$. We also prove the abundance conjecture in dimension $\leq 5$ when $\kappa\geq 0$ and $\nu\leq 1$ unconditionally.