Relative $\dbar$-complex and its curvature properties

TL;DR

This paper introduces the relative ar-complex, explores its curvature properties, and connects Yamaguchi's subharmonicity theory of the Green operator to curvature of quotient bundles, with additional recent applications discussed.

Contribution

It defines the relative ar-complex and links its curvature properties to subharmonicity theories, providing new geometric insights and applications.

Findings

Curvature properties of the relative ar-complex are characterized.

Yamaguchi's subharmonicity of the Green operator is interpreted as a curvature property.

Survey of recent applications of these curvature concepts.

Abstract

We shall define the relative $\dbar$-complex and study the curvature properties of the associated vector bundles. As an application, we shall prove that Yamaguchi's theory on subharmonicity of the Green operator can be seen as a curvature property of the quotient bundle. A short survey of other recent applications will also be given in this paper.