Nonexistence of decreasing equisingular approximations with logarithmic poles

TL;DR

This paper proves that for complex manifolds of dimension greater than one, certain multiplier ideal sheaves lack decreasing equisingular approximations with logarithmic poles, highlighting a fundamental limitation in complex analysis.

Contribution

It demonstrates the nonexistence of decreasing equisingular approximations with logarithmic poles for specific multiplier ideal sheaves on higher-dimensional complex manifolds.

Findings

Existence of multiplier ideal sheaves without equisingular logarithmic pole approximations

Nonexistence of decreasing equisingular approximations in higher dimensions

Highlights limitations in complex analytic approximation methods

Abstract

In this article, we present that for any complex manifold whose dimension is bigger than one, there exists a multiplier ideal sheaf such that there don't exist equisingular weights with logarithmic poles, which are not smaller than the orginal weight. A direct consequence is the nonexistence of decreasing equisingular approximations with logarithmic poles.