Singular metrics and a conjecture by Campana and Peternell

TL;DR

This paper explores a conjecture linking the Kodaira and Iitaka dimensions in algebraic geometry, showing its near equivalence to the non-vanishing conjecture using recent advances in singular metrics.

Contribution

It demonstrates that the Campana-Peternell conjecture is nearly equivalent to the non-vanishing conjecture, connecting two significant conjectures in algebraic geometry.

Findings

Campana-Peternell conjecture is almost equivalent to the non-vanishing conjecture.

Recent work on singular metrics on pluri-adjoint bundles supports this equivalence.

The results provide a new perspective on the relationship between Kodaira and Iitaka dimensions.

Abstract

A conjecture by Campana and Peternell says that if a positive multiple of $K_X$ is linearly equivalent to an effective divisor $D$ plus a pseudo-effective divisor, then the Kodaira dimension of $X$ should be at least as big as the Iitaka dimension of $D$. This is a very useful generalization of the non-vanishing conjecture (which is the case $D = 0$). We use recent work about singular metrics on pluri-adjoint bundles to show that the Campana-Peternell conjecture is almost equivalent to the non-vanishing conjecture.