A smoothness criterion for complex spaces in terms of differential forms

For a reduced pure dimensional complex space $X$, we show that if Barlet's recently introduced sheaf $α_X^1$ 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 $ω_X^1$.

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