Approximations and examples of singular Hermitian metrics on vector bundles

TL;DR

This paper investigates singular Hermitian metrics on vector bundles, establishing coherence results for higher rank multiplier ideals and classifying negatively curved metrics on specific bundles, advancing understanding in complex geometry.

Contribution

It introduces new coherence results for higher rank multiplier ideals and classifies negatively curved metrics on certain vector bundles, extending previous theories.

Findings

Coherence of higher rank multiplier ideals established.

Positively curved metrics with non-Nakano semipositive approximations analyzed.

Complete classification of negatively curved metrics on specific bundles achieved.

Abstract

We study singular Hermitian metrics on vector bundles. There are two main results in this paper. The first one is on the coherence of the higher rank analogue of multiplier ideals for singular Hermitian metrics defined by global sections. As an application, we show the coherence of the multiplier ideal of some positively curved singular Hermitian metrics whose standard approximations are not Nakano semipositive. The aim of the second main result is to determine all negatively curved singular Hermitian metrics on certain type of vector bundles, for example, certain rank 2 bundles on elliptic curves.