Given a complete algebraic variety $X$ over $\mathbb{C}$ and a vector bundle $E$ of rank $r$, let $\Omega(X,E)$ denote the graded ring $\bigoplus_{m\ge 0}H^0(X,S^mE)$, and define $$\lambda(E,X)=\operatorname{tr.deg}\Omega(X,E)-r$$ to be the $E$-dimension of $X$. As part of his $\lambda$-classification of algebraic surfaces, Sakai (MR0555717) shows that those with $\lambda (X):=\lambda(\Omega_X, X) =2$ are necessarily of general type ($\kappa =2$). Thus, any smooth surface $X$ with big $\Omega_X$ is necessarily of general type.

Could anyone point out other references that prove this last fact, or discuss the analogous case in higher dimensions?