BMY inequality for surfaces of general type in characteristic 0

Question

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?

Answer 1

Every property you describe is preserved by base change from one algebraically closed field to another, so by the Lefschetz principle it works the same over an arbitrary algebraically closed field of characteristic zero as over the complex numbers.