Title: Mapping properties of the Hilbert and Fubini--Study maps in Kähler geometry,
Abstract: Suppose that we have a compact Kähler 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.