Toward a higher codimensional Ueda theory

TL;DR

This paper extends Ueda's flatness theory from hypersurfaces to codimension two, providing new criteria for the anti-canonical bundle's properties on blow-ups of projective space.

Contribution

It introduces a codimension two analogue of Ueda's theory and applies it to analyze the semi-ampleness of anti-canonical bundles on specific blow-ups.

Findings

Anti-canonical bundle of the blow-up at 8 points is non semi-ample.

Existence of a smooth Hermitian metric with semi-positive curvature.

New criteria for flatness in higher codimension.

Abstract

Ueda's theory is a theory on a flatness criterion around a smooth hypersurface of a certain type of topologically trivial holomorphic line bundles. We propose a codimension two analogue of Ueda's theory. As an application, we give a sufficient condition for the anti-canonical bundle of the blow-up of the three dimensional projective space at $8$ points to be non semi-ample however admit a smooth Hermitian metric with semi-positive curvature.