In complex dimension $2$, if a surface $S$ is a blowup of a surface $S'$, one has the following relation between their Chern numbers :
$c_1^2(S) + 1 = c_1^2(S')$
$c_2(S) - 1 = c_2(S')$
By using this fact one gets : for the Chern numbers $c_1^2$ and $c_2$ of the tangent bundle of a compact complex surface $S$ to be zero implies for $S$ to be minimal, and to belong to only some specific families among those listed in the Enriques-Kodaira classification according to this table for example : https://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification#/media/File:Geography_of_surfaces.jpg
And if the Chern numbers are both negative, then the surface has to be either ruled or a blowup of a ruled surface.
My question is : are there similar conclusions one could draw in bigger dimension ?