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Quantitative injectivity of the Fubini--Study map

Yoshinori Hashimoto

TL;DR

This work provides a quantitative version of the injectivity of the Fubini-Study map for polarized varieties, showing that the distance between Kodaira embedding data $A$ and $B$ is controlled polynomially in the degree $k$ by a Sobolev norm of the discrepancy function $f_k(A,B;H_k)$, with the bound $\|A^{-1}-B^{-1}\|^2_{HS(H_k)} \le C_h k^{n} \|f_k(A,B;H_k)\|^2_{W^{2,2}(\omega_h)}$. The proof blends Bergman kernel asymptotics, positivity of the second fundamental form, and Lie-algebra decompositions to relate matrix differences to Hamiltonian data on projective space. The paper also rectifies previous arguments (notably in yhhilb and related works) and extends the framework to handle subtle issues surrounding the choice of reference metric. Overall, the result advances quantitative geometric understanding of projective embeddings and provides tools for stability analyses in complex differential geometry.

Abstract

We prove a quantitative version of the injectivity of the Fubini--Study map that is polynomial in the exponent of the ample line bundle, and correct the arguments in the author's previous papers.

Quantitative injectivity of the Fubini--Study map

TL;DR

This work provides a quantitative version of the injectivity of the Fubini-Study map for polarized varieties, showing that the distance between Kodaira embedding data and is controlled polynomially in the degree by a Sobolev norm of the discrepancy function , with the bound . The proof blends Bergman kernel asymptotics, positivity of the second fundamental form, and Lie-algebra decompositions to relate matrix differences to Hamiltonian data on projective space. The paper also rectifies previous arguments (notably in yhhilb and related works) and extends the framework to handle subtle issues surrounding the choice of reference metric. Overall, the result advances quantitative geometric understanding of projective embeddings and provides tools for stability analyses in complex differential geometry.

Abstract

We prove a quantitative version of the injectivity of the Fubini--Study map that is polynomial in the exponent of the ample line bundle, and correct the arguments in the author's previous papers.

Paper Structure

This paper contains 3 sections, 3 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['ppqifs']}
  3. Correction to yhextremal

Key Result

Theorem 1.2

There exist constants $k_0 \in \mathbb{N}$ and $C_h >0$, depending only on $h$, such that we have for any $A, B \in \mathcal{B}_k$ and for any $k \ge k_0$, where $\Vert \cdot \Vert_{\mathrm{HS}(H_k)}$ is the Hilbert--Schmidt norm with respect to $H_k = \mathrm{Hilb}(h^k)$ and $\Vert \cdot \Vert_{W^{2,2} (\omega_h)}$ is the Sobolev norm with respect to $\omega_h$.

Theorems & Definitions (7)

  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['ppqifs']}
  • Remark 2.3