Let $X$ be a smooth projective variety over $\mathbb{C}$. For an ample divisor H, we can define the slop of vector bundle with respect to $H$, then we can define stablilty of vector bundle with respect to $H$. However it might be possible that an $H$-stable vector bundle is no longer stable with respect to another ample divisor $H'$.  

 On the other hand the famous theory of Uhlenbeck-Yau about irreducible representation and stable bundle implies that if a vector bundle has trivial Chern classes, then its stablility does not depend on the ample divisor. I wonder is there a direct proof of this fact without using Uhlenbeck-Yau thoery?


A natural question to ask : does the statement hold in charateristic p case ( one may need some condition like algebraic closed field, the variety has $W\_2$-lifting, etc.)? 



The stability of vector bundle with trivial Chern classes is independent of ample divisor, a direct proof?
See the following paper by Adrian Langer: <http://arxiv.org/abs/0905.4600> and more precisely section 4.


I quote:



> 
> In this section we show that strongly semistable torsion free sheaves with vanishing Chern classes
>  are locally free and that they are strongly semistable with respect to all polarizations
> 
> 
> 


(A torsion free sheaf $\mathcal{E}$ on $X$ is called strongly semistable if it is semistable and if all ${F^e}^\ast \mathcal{E}$ remain semistable, where $F: X \to X$ is the absolute Frobenius.)


More precisely Proposition 4.5 of that paper should prove what you want. Langer works in arbitrary characteristics (over an algebraically closed field).


