Uniqueness of irreducible desingularization of singularities associated to negative vector bundles

TL;DR

This paper proves the uniqueness of irreducible desingularizations of certain singularities related to negative vector bundles and explores their implications for the isomorphism of negative line bundles and modifications of submanifolds.

Contribution

It establishes the uniqueness of desingularizations for Grauert blow downs of negative holomorphic vector bundles and characterizes when negative line bundles are isomorphic based on their singularities.

Findings

Irreducible desingularizations are unique up to isomorphism.

Negative line bundles are isomorphic iff their Grauert blow downs have isomorphic germs.

The blow-up along a submanifold provides a unique modification to a hypersurface.

Abstract

We prove that the irreducible desingularization of a singularity given by the Grauert blow down of a negative holomorphic vector bundle over a compact complex manifold is unique up to isomorphism, and as an application, we show that two negative line bundles over compact complex manifolds are isomorphic if and only if their Grauert blow downs have isomorphic germs near the singularities. We also show that there is a unique way to modify a submanifold of a complex manifold to a hypersurface, namely, the blow up of the ambient manifold along the submanifold.