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Nakano positivity of singular Hermitian metrics: Approximations and applications

Takahiro Inayama, Shin-ichi Matsumura

Abstract

This paper studies the approximation of singular Hermitian metrics on vector bundles using smooth Hermitian metrics with Nakano semi-positive curvature on Zariski open sets. We show that singular Hermitian metrics capable of this approximation satisfy Nakano semi-positivity as defined through the $\overline{\partial} $-equation with optimal $L^2$-estimates. Furthermore, for a projective fibration $f \colon X \to Y$ with a line bundle $L$ on $X$, we provide a specific condition under which the Narasimhan-Simha metric on the direct image sheaf $f_{*}\mathcal{O}_{X}(K_{X/Y}+L)$ admits this approximation. As an application, we establish several vanishing theorems.

Nakano positivity of singular Hermitian metrics: Approximations and applications

Abstract

This paper studies the approximation of singular Hermitian metrics on vector bundles using smooth Hermitian metrics with Nakano semi-positive curvature on Zariski open sets. We show that singular Hermitian metrics capable of this approximation satisfy Nakano semi-positivity as defined through the -equation with optimal -estimates. Furthermore, for a projective fibration with a line bundle on , we provide a specific condition under which the Narasimhan-Simha metric on the direct image sheaf admits this approximation. As an application, we establish several vanishing theorems.
Paper Structure (5 sections, 12 theorems, 53 equations)

This paper contains 5 sections, 12 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Definitions of Nakano positivity and preliminary results
  3. Proof of Theorem \ref{['thm-seqimpliesL2']}
  4. Applications of Theorem \ref{['thm-seqimpliesL2']}
  5. On the positivity of direct image sheaves

Key Result

Theorem 1.1

Let $f \colon X \to Y$ be a projective fibration between $($not necessarily compact$)$ complex manifolds $(X, \omega_{X})$, $(Y, \omega_{Y})$ with Hermitian forms. Let $\theta$ be a continuous real $(1,1)$-form on $Y$ and $L$ be a line bundle on $X$ satisfying the following conditions$:$ Then, the induced Narasimhan-Simha metric $G$ on the direct image sheaf $f_{*} \mathcal{O}_{X}(K_{X/Y}+L)$ sati

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Associated Hermitian forms and Nakano/Griffiths positivity
  • Definition 2.2
  • Definition 2.3: Griffiths positivity
  • Definition 2.4: Nakano positivity in the sense of approximations
  • Remark 2.5
  • Definition 2.6: Ina22
  • ...and 17 more