I know a definition of torsion free sheaf $\mathcal{F}$ on an integral scheme X, That is: $\mathcal F$ is an quasi-coherent sheaf such that it hasn't have not-null torsion sections, which is equivalent to $\forall U$ affine open subset $\mathcal F(U)$ is a torsion free $\mathcal O(U)$-module Torsion free modules.
Does a generalization for torsion free sheafs on complex manifolds exists? Which is it?