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.

8 pages. Previously part of arXiv:1508.02643v1, separated as an independent paper with a new result added; v2. 15 pages, exposition significantly improved. To appear in Ann. Fac. Sci. Toulouse Math; v3. erratum added, see also arXiv:2502.08038