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arXiv · 2402.06883

Nakano positivity of singular Hermitian metrics: Approximations and applications

Takahiro Inayama·Shin-ichi Matsumura
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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.

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Takahiro Inayama, Shin-ichi Matsumura. 2024-02-10. Nakano positivity of singular Hermitian metrics: Approximations and applications. https://arxiv.org/abs/2402.06883

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