Mapping properties of the Hilbert and Fubini--Study maps in Kähler geometry

TL;DR

The paper investigates two central maps in polarized Kähler geometry: the Hilbert map $\mathrm{Hilb}$ and the Fubini--Study map $FS$, relating Hermitian forms on $H^0(X,\mathcal{L})$ to Hermitian metrics on $\mathcal{L}$. It originally asserts that for a very ample line bundle $\mathcal{L}$, $\mathrm{Hilb}$ is surjective and $FS$ is injective, with a degree-theoretic approach connected to the Aubin--Yau theorem. The erratum shows that the surjectivity of $\mathrm{Hilb}$ is false (with counterexamples), while the injectivity of $FS$ remains true, though the paper’s own proof is invalid; independent work by Lempert provides a correct proof and a refined description of the image. The authors develop a robust finite-dimensional framework using the maps $Q_{\mathbb{P}}$ and $Q_X$, analyze boundary behavior via compactifications and Hilbert–Chow limits, and extend the discussion to variants of $\mathrm{Hilb}$ for different volume forms. These results clarify when finite-dimensional balancing procedures can realize prescribed inner products on $H^0(X,\mathcal{L})$ and have implications for stability and quantization 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.

Theorems & Definitions (30)

• Theorem 1.1
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Theorem 2.4
• Theorem 2.5
• Definition 2.6
• Lemma 2.7
• proof
• Lemma 2.8