Local and Global Heights on Weighted Projective Varieties

TL;DR

This paper extends the theory of weighted heights from weighted projective spaces to weighted varieties and subvarieties, establishing foundational properties and metrics for these geometric objects.

Contribution

It introduces a comprehensive framework for local and global weighted heights on weighted varieties, including the existence of locally bounded weighted metrics for line bundles.

Findings

Defined local and global weighted heights for weighted varieties and subvarieties.

Proved that any line bundle on a weighted variety admits a locally bounded weighted metric.

Extended height theory to more general weighted algebraic structures.

Abstract

We investigate local and global weighted heights a-la Weil for weighted projective spaces via Cartier and Weil divisors and extend the definition of weighted heights on weighted projective spaces from arXiv:1902.06563 to weighted varieties and closed subvarieties. We prove that any line bundle on a weighted variety admits a locally bounded weighted $M$-metric. Using this fact, we define local and global weighted heights for weighted varieties in weighted projective spaces and their closed subschemes and show their fundamental properties.