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?

  • $\begingroup$ What is your definition of a big vector bundle? Is the determinant of a big vector bundle (with your definition) not big? $\endgroup$ Jun 22 at 12:52
  • $\begingroup$ @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. $\endgroup$
    – astana
    Jun 22 at 14:02
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    $\begingroup$ 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. $\endgroup$ Jun 22 at 18:27
  • $\begingroup$ @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. $\endgroup$
    – astana
    Jun 26 at 17:27


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