Hermitian metrics on the anti-canonical bundle of the blow-up of the projective plane at nine points

TL;DR

This paper studies Hermitian metrics on the anti-canonical bundle of a rational surface formed by blowing up the projective plane at nine points, using deformation of complex structures to analyze their properties.

Contribution

It introduces a modified approach based on Ueda's argument to examine the complex analytic structure near subvarieties in this context.

Findings

Analysis of Hermitian metrics on the anti-canonical bundle

Application of deformation techniques to complex structures

Insights into the neighborhood structure of the blown-up surface

Abstract

We investigate Hermitian metrics on the anti-canonical bundle of a rational surface obtained by blowing up the projective plane at nine points. For that purpose, we pose a modified variant of an argument made by Ueda on the complex analytic structure of a neighborhood of a subvariety by considering the deformation of the complex structure.