Curvature operator of holomorphic vector bundles and $L^2$-estimate condition for $(n,q)$ and $(p,n)$-forms

TL;DR

This paper investigates the positivity properties of the curvature operator in holomorphic Hermitian vector bundles, providing new characterizations of semi-positivity and semi-negativity for specific forms using L2-estimates.

Contribution

It introduces novel characterizations of semi-positive and semi-negative curvature operators for certain forms, extending known results on Nakano semi-positivity.

Findings

New criteria for semi-positive curvature operators for (n,q) and (p,n)-forms

Characterizations of Nakano semi-negativity via L2-estimates

Enhanced understanding of curvature operator properties in complex geometry

Abstract

We study the positivity properties of the curvature operator for holomorphic Hermitian vector bundles. We obtain new characterization of semi-positive curvature operators for $(n,q)$ and $(p,n)$-forms by L2-estimates. The characterization of Nakano semi-positivity by $L^2$-estimate is already known. Applying our results, we give new characterizations of Nakano semi-negativity.