# Mapping properties of the Hilbert and Fubini--Study maps in K\"ahler geometry

## TL;DR

This paper explores the relationship between hermitian forms on sections of a line bundle over a Kähler manifold and the associated Fubini--Study map, proving its injectivity and characterizing the mapping properties in Kähler geometry.

## Contribution

It establishes that any positive hermitian form on sections can be realized via an $L^2$-inner product, and proves the injectivity of the Fubini--Study map in this setting.

## Key findings

- Any positive definite hermitian form arises from an $L^2$-inner product.
- The Fubini--Study map is injective.
- Mapping properties of the Fubini--Study map are characterized.

## Abstract

Suppose that we have a compact K\"ahler manifold $X$ with a very ample line bundle $\mathcal{L}$. We prove that any positive definite hermitian form on the space $H^0 (X,\mathcal{L})$ of holomorphic sections can be written as an $L^2$-inner product with respect to an appropriate hermitian metric on $\mathcal{L}$. We apply this result to show that the Fubini--Study map, which associates a hermitian metric on $\mathcal{L}$ to a hermitian form on $H^0 (X,\mathcal{L})$, is injective.