Grauert-Riemenschneider multiplier ideal sheaves and the (optimal) Brian\c{c}on-Skoda number

TL;DR

This paper surveys recent results on Grauert-Riemenschneider multiplier ideal sheaves on complex spaces and determines the Briançon-Skoda number for certain Noetherian rings with weakly holomorphic functions, addressing a question by Huneke.

Contribution

It provides the first computation of the Briançon-Skoda number for a broad class of Noetherian rings with weakly holomorphic functions and weakly rational singularities.

Findings

Determined the Briançon-Skoda number for rings with weakly rational singularities.

Extended the understanding of multiplier ideal sheaves on complex spaces.

Partially answered a question posed by Huneke.

Abstract

The goal of this note is to survey some recent results on the Grauert-Riemenschneider multiplier ideal sheaves on any (reduced) complex space of pure dimension. In particular, we obtain the Brian\c{c}on-Skoda number for any Noetherian ring of weakly holomorphic functions with weakly rational singularities (\emph{not} essentially of finite type over $\mathbb{C}$ and Cohen-Macaulay local rings), which will partially answer a question of Huneke.