A smoothness criterion for complex spaces in terms of differential forms

TL;DR

This paper establishes a criterion for smoothness of complex spaces based on the local freeness of certain sheaves of differential forms, linking algebraic properties to geometric smoothness.

Contribution

It introduces a new smoothness criterion for complex spaces using the local freeness of sheaves of holomorphic and weakly holomorphic 1-forms.

Findings

Local freeness of sheaf $eta_X^1$ implies smoothness of $X$

Connection established between sheaves $eta_X^1$ and $ar{eta}_X^1$

Provides a criterion linking differential forms to complex space smoothness

Abstract

For a reduced pure dimensional complex space $X$, we show that if Barlet's recently introduced sheaf $\alpha_X^1$ of holomorphic $1$-forms or the sheaf of germs of weakly holomorphic $1$-forms is locally free, then $X$ is smooth. Moreover, we discuss the connection to Barlet's well-known sheaf $\omega_X^1$.