Nakano positivity of singular Hermitian metrics: Approximations and applications

2024/02/10 by Takahiro Inayama, Inayama, Takahiro, Shin‐ichi Matsumura +1
Mathematics · Physics and Astronomy · #32L20 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Primary 32U05 #Quantum Mechanics and Non-Hermitian Physics #Secondary 32A70

paper · pdf · doi:10.48550/arxiv.2402.06883

openalex publication_date 2024/02/10 · openalex created_date 2024/02/14 · openalex updated_date 2026/07/28

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 L²-estimates. Furthermore, for a projective fibration f \colon X → 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_*O_X(K_X/Y+L) admits this approximation. As an application, we establish several vanishing theorems.

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