Subsets of full measure in a generic submanifold in $\C^n$ are non-plurithin

TL;DR

This paper proves that in a smooth generic submanifold of complex space, the complement of a measure-zero subset is non-plurithin at every point, extending previous results to broader conditions with a new proof approach.

Contribution

It generalizes previous work by showing the complement of measure-zero sets in smooth submanifolds is non-plurithin, using a novel proof technique.

Findings

Complement of measure-zero sets in smooth submanifolds is non-plurithin.

Extends previous results from pluripolar sets to measure-zero sets.

Provides a new proof method for non-plurithinness in complex submanifolds.

Abstract

In this paper we prove that if $I $ is a subset of measure 0 in a $C^2-$smooth generic submanifold $M$ of $ \C^n$, then its complement in $M$ is non-plurithin at each point of $M$ in $\C^n$. This result improves a previous result of A. Edigarian and J. Wiegerinck, who considered the case where $I$ is pluripolar set contained in a $C^1-$smooth generic submanifold $M \subset \C^n$. The proof of our result is essentially different.