# Surfaces with $\Omega_X$ big are of general type

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?

• What is your definition of a big vector bundle? Is the determinant of a big vector bundle (with your definition) not big? Jun 22 at 12:52
• @AriyanJavanpeykar I am using the notion $L$-bigness, i.e., $E$ is big iff the tautological line bundle of $\mathbb{P}(E)$ is big. One usually requires global generation or nefness of $E$ to insure that $E$ big implies $\det(E)$ big. Jun 22 at 14:02
• I see. This is what is sometimes called "big in the sense of Hartshorne". If I'm reading things right, then Prop. 5.3 of arxiv.org/abs/2206.15399v2 it is claims that whenever $\Omega_X$ is big in the sense of Hartshorne, then its determinant is big. Jun 22 at 18:27
• @AriyanJavanpeykar Thanks! That's helpful. I found that Brunebarbe in an earlier paper of his concludes (Theorem 2.9) that any complex algebraic variety with big cotangent bundle (equiv. maximal cotangent dimension) is of general type. Jun 26 at 17:27