Sign of first chern class with some conditions

Question

Let $X$ be a compact Kahler algebraic variety which $K_M$ is big and nef, and $Kod(X)=dimX$ then why the first chern class $c_1(M)$ is negative or zero . I don't undrestand kawamata's theorem in this case

Answer 1

I don't know which theorem of Kawamata you are referring to, but that doesn't seem to matter for this question.

The important point to understand here is that $K_M$ is the determinant line bundle of the cotangent bundle $\Omega_M$, which is the dual of the tangent bundle $T_M$.

So $$c_1(K_M)=c_1(\Omega_M)=-c_1(T_M).$$

So if $K_M$ is nef, meaning that it has degree $\geq 0$ on every curve, then $c_1(T_M)$ has degree $\le 0$ on every curve.