Is it true that if $F$ is a $\mu$-semistable sheaf, then its double dual is also $\mu$-semistable?
We know $c_1(F)=c_1(F^{**})$. Given a sub sheaf $E$ of $F^{**}$, if we can associate to it some subsheaf of $F$, then we can make use of the $\mu$-semistability of $F$. But I am stuck here.
Thanks for the help!
There is a canonical map $F \to F^{**}$, which is injective when $F$ is torsion-free. So, given a subsheaf $E \subset F^{**}$ you can always consider its premiage in $F$, i.e., $$ E' = Ker(E \to Coker(F \to F^{**})). $$ If $F$ is torsion-free, then $\mu(F) = \mu(F^{**})$ and $\mu(E') = \mu(E)$ (since the sheaves are isomorphic on a complement of a subset of codimension 2).