Ueda theory for compact curves with nodes

TL;DR

This paper extends Ueda's theory to compact complex curves with nodes, analyzing neighborhoods and applying the results to singular Hermitian metrics on blow-ups of the projective plane.

Contribution

It introduces an analogue of Ueda's theory for nodal curves and explores applications to metrics on anti-canonical bundles.

Findings

Extended Ueda theory to curves with nodes

Analyzed neighborhoods of nodal curves

Applied to semi-positive curvature metrics on blow-ups

Abstract

Let $C$ be a compact complex curve included in a non-singular complex surface such that the normal bundle is topologically trivial. Ueda studied complex analytic properties of a neighborhood of $C$ when $C$ is non-singular or is a rational curve with a node. We propose an analogue of Ueda's theory for the case where $C$ admits nodes. As an application, we study singular Hermitian metrics with semi-positive curvature on the anti-canonical bundle of the blow-up of the projective plane at nine points in arbitrary position.