Minimal singular metrics of a line bundle admitting no Zariski-decomposition

TL;DR

This paper provides an explicit form of minimal singular metrics for certain big line bundles on complex manifolds where Zariski-decomposition fails, and explores related geometric properties.

Contribution

It offers a concrete expression for minimal singular metrics on line bundles lacking Zariski-decomposition, expanding understanding of their geometric and analytic properties.

Findings

Explicit minimal singular metrics constructed

Zariski-closedness of non-nef loci analyzed

Openness conjecture discussed in this context

Abstract

We give a concrete expression of a minimal singular metric of a big line bundle on a compact K\"ahler manifold which is the total space of a toric bundle over a complex torus. In this class of manifolds, Nakayama constructed examples which have line bundles admitting no Zariski-decomposition even after any proper modifications. As an application, we discuss the Zariski-closedness of non-nef loci and the openness conjecture of Demailly and Koll\'{a}r in this class.