Let $X$ be a smooth, complex, projective, minimal surface of general type, i.e. the canonical (line) bundle $K_X$ is big and nef.

It is known that $3c_2\geq c_1^2$ (the Bogomolov-Miyaoka-Yau inequality) and if the equality holds then $K_X$ is ample, where for $k\in\{1,2\},\,c_k\equiv c_k(T_X)$ are the Chern classes of the tangent bundle.

Questions. Let $X$ be a smooth, projective surface defined over an algebraically closed field $\mathbb{F}$ of characteristic $0$.

1. If $K_X$ big and nef, does the BMY inequality hold?
2. Otherwise, what are the right hypothesis on $X$ under which the BMY inequality holds?
3. If $K_X$ big (and nef), could one define $X$ (minimal) surface of general type?