An ampleness criterion with the extendability of singular positive metrics

TL;DR

This paper establishes a new criterion linking the extendability of singular positive metrics along subvarieties to the ampleness of line bundles on projective manifolds, providing a converse to a known extension result.

Contribution

It proves that the ability to extend singular positive metrics along subvarieties characterizes the ampleness of line bundles, offering a new criterion for ampleness.

Findings

Extendability of singular positive metrics implies ampleness.

Provides a converse to the known extension theorem by Coman, Guedj, and Zeriahi.

Characterizes ampleness via metric extendability.

Abstract

Coman, Guedj and Zeriahi proved that, for an ample line bundle $L$ on a projective manifold $X$, any singular positive metric on the line bundle $L|_{V}$ along a subvariety $V \subset X$ can be extended to a global singular positive metric on $L$. In this paper, we prove that the extendability of singular positive metrics on a line bundle along a subvariety implies the ampleness of the line bundle.