Multiplicites and modifications, and singularities associated to blowing down negative vector bundles

TL;DR

This paper explores the relationship between multiplicities, modifications, and singularities in complex geometry, providing formulas and estimates for singularities arising from blow-downs of negative vector bundles.

Contribution

It introduces a new interpretation of mixed Hilbert-Samuel multiplicities as intersection numbers and applies this to analyze singularities from Grauert blow-downs.

Findings

Generalized Lelong numbers for multiplicities

Exact formulas for singularities from negative vector bundle blow-downs

Estimates for multiplicities of isolated singularities

Abstract

We first present the mixed Hilbert-Samuel multiplicities of analytic local rings over \mathbb{C} as generalized Lelong numbers and further represent them as intersection numbers in the context of modifications. As applications, we give estimates or an exact formula for the multiplicities of isolated singularities that given by the Grauert blow-downs of negative holomorphic vector bundles.