Let $(X, mathcal{O}_X)$ be a complex manifold, which we can take to be projective. A coherent subsheaf $mathscr{F}$ of some sheaf $mathscr{G}$ is said to be *saturated* in $mathscr{G}$ if the quotient sheaf $mathscr{G}/mathscr{F}$ is torsion-free. Further, we can define the *saturation* of $mathscr{F}$ inside $mathscr{G}$ to be the kernel of the map $$mathscr{G} to (mathscr{G}/mathscr{F})/(text{torsion}).$$

What is the intuition for the saturation of a subsheaf? Are there some elementary intuition-granting examples?

**Example**: Let $E$ be an elliptic curve and $C$ a hyper elliptic curve with an involution $sigma$ such that the quotient $hat{C} := C/sigma$ is $mathbb{P}^1$. Take $tau$ to be a translation of order $2$ on $E$ and define $X = (C times E)/(sigma, tau)$. Let $f : X to C’$ be the Iitaka fibration of $X$, and let $mathscr{L} = f^{ast}K_{C’}$, which is a subsheaf of $Omega_X^1$. The saturation is then the kernel of $$Omega_X^1 to (Omega_X^1 / mathscr{L}) / text{(torsion)}.$$

Can we get an explicit description of the saturation in this case?