# Stable torsion free sheaf on smooth projective surface

Let $E$ be a torsion-free sheaf on a smooth projective variety $X$ over $\mathbb{C}$. Let $H$ be an ample line bundle on $X$. Then we say $E$ is stable if $\mu_{H}(F)<\mu_{H}(E),\,\forall 0\neq F \subset E,$ (where slope of torsion free sheaf $F$ is defined by $\mu_{H}(F)=\frac{c_1(F)\cdot H}{rk(F)}$.)

I feel when $X$ is a smooth projective surface, $E$ is stable if and only if $\mu_{H}(F)<\mu_{H}(E),\forall 0 \neq F \subset E$, (where $F$ is a vector bundle). This is because $F^{**}$ is vector bundle on the surface $X$ with $\mu_{H}(F^{**})=\mu_{H}(F)$.

Question: 1) Is the above reason correct?

2) For higher-dimensional smooth projective variety(>2) can we say $E$ is stable if and only if $\mu_{H}(F)<\mu_{H}(E),\forall 0 \neq F \subset E$, (where $F$ is a vector bundle)

I.e. is it enough to consider $F$ a vector bundle to define stability? (The same logic does not go through in this case, since $F^{**}$ is not locally free for any given torsion free sheaf $F$).

The above statement on surfaces is correct if you suppose $$E$$ to be locally free, otherwise you should pay attention to torsion free subsheaves as well. I am thinking about the direct sum $$E = I_p \oplus \mathcal{O} (-1)$$ of the ideal sheaf $$I_p$$ of a point and the line bundle $$\mathcal{O} (-1)$$ on the projective plane. Then $$E$$ is torsion free, and should have no subsheaf which is torsion free and of higher slope, however it does have the subsheaf $$I_p$$ which destabilizes it.
In the same way, on a higher dimensional variety, the condition (2) is correct if you suppose that $$E$$ and $$F$$ are reflexive sheaves (i.e. they are isomorphic to their bidual sheaves), which for surfaces is the same as being locally free.