# Norming sets on a compact complex manifold

## TL;DR

This paper characterizes norming sets for spaces of global holomorphic sections of powers of positive line bundles on compact complex manifolds, providing metric conditions for these sets to control the sections' norms.

## Contribution

It offers a metric characterization of norming sets for holomorphic sections of line bundle powers on compact complex manifolds, extending understanding of their geometric properties.

## Key findings

- Provides a criterion for norming sets in terms of measurable subsets.
- Establishes bounds relating global sections and integrals over these subsets.
- Characterizes the sequences of subsets that serve as norming sets for all sections.

## Abstract

We describe the norming sets for the space of global holomorphic sections to a $k$-power of a positive holomorphic line bundle on a compact complex manifold $X$. We characterize in metric terms the sequence of measurable subsets $\{G_{k}\}_{k}$ of $X$ such that there is a constant $C > 0$ where $$\|s\|^{2}\leq C \int_{G_{k}} |s(z)|^{2}\ dV(z)$$ for every $s\in H^{0}(L^{k})$ and for all $k\in\mathbb{N}$.