On Generalised Abundance, II

TL;DR

This paper advances the understanding of the Generalised Nonvanishing Conjecture in algebraic geometry, providing new nonvanishing results for pluricanonical bundles and progress towards the conjecture across all dimensions.

Contribution

It establishes that a stronger form of the Generalised Nonvanishing Conjecture holds in most cases with certain metrics, assuming the MMP in lower dimensions.

Findings

Derived new nonvanishing results for pluricanonical bundles.

Progressed towards the Generalised Nonvanishing Conjecture in all dimensions.

Showed that a stronger version of the conjecture holds under specific metric conditions.

Abstract

In our previous work, we introduced the Generalised Nonvanishing Conjecture, which generalises several central conjectures in algebraic geometry. In this paper, we derive some surprising nonvanishing results for pluricanonical bundles which were not predicted by the Minimal Model Program, by making progress towards the Generalised Nonvanishing Conjecture in every dimension. The main step is to establish that a somewhat stronger version of the Generalised Nonvanishing Conjecture holds almost always in the presence of metrics with generalised algebraic singularities, assuming the Minimal Model Program in lower dimensions.