Fumio Takemoto, Stable vector bundles on algebaric surfaces, Proposition 2.7 says that,
Paper: http://projecteuclid.org/euclid.nmj/1118798682
Let $E$ be a vector bundle of rank two. If $N(E)=c_1^2(E)-4c_2(E)>0$, then $E$ is $H$ stable if and only if $E$ is $H'$ stable for any ample line bundle $H'$ on $X$.
My doubt is this, Bogomolov's Inequality says that,
Let $X$ be an algebraic surface and let $H$ be an ample divisor on $X$. Suppose that $E$ is an $H$-stable rank $2$ vector bundle on $X$. Then $N(E)\leq 0$.
Which means condition in Proposition 2.7 not possible. I am misunderstanding something. Please correct me.
Also in same paper Proposition 2.10 says that,
Let $X$ be a surface and $E$ an $H$-semi-stable vector bundle on $X$ with $d(E, H) = 0$. Then $dim H^0(E)\leq rk E$. And the equality holds if and only if $E$ is free.
Vector bundle is always locally free. What does it mean to be free in this prop?