# Canonical singular hermitian metrics on relative canonical bundles

## TL;DR

This paper introduces a new class of canonical metrics called supercanonical AZD's on canonical bundles of smooth projective varieties with pseudoeffective canonical classes, and studies their behavior under deformations.

## Contribution

It defines supercanonical AZD's and proves their invariance properties, providing new insights into the variation of canonical metrics and plurigenera.

## Key findings

- Introduction of supercanonical AZD's on canonical bundles.
- Proof of invariance of plurigenera using these metrics.
- Analysis of metric variation under projective deformations.

## Abstract

We introduce a new class of canonical AZD's (called the supercanonical AZD's) on the canonical bundles of smooth projective varieties with pseudoeffective canonical classes. We study the variation of the supercanonical AZD $\hat{h}_{can}$ under projective deformations and give a new proof of the invariance of plurigenera.