# ag.algebraic geometry – Saturation of sheaves

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?