Frobenius pull-back of vector bundles of rank 2 over non-uniruled varieties
In positive characteristic, semi-stability of vector bundles is not preserved under surjective morphisms. This makes it impossibility to prove, in positive characteristic, important results such as the vanishing theorem of Kodaira-Mumford or the Bogomolov-Gieseker inequality in the same way one does in characteristic 0. However, \textit{T. Ekedahl} [Publ. Math., Inst. Hautes Étud. Sci. 67, 97-144 (1988; Zbl 0674.14028)] and \textit{N. I. Shepherd-Barron} [Invent. Math. 106, 243-262 (1991; Zbl 0769.14006)] observed, in the surface case, that the break down of semi-stability under inseparable morphisms might uncover intriguing information on the variety in question. In this paper, the author explains these results in terms of a simple and general principle: if a Frobenius pull-back of a rank 2 vector bundle is very unstable then the variety is uniruled. His results are consequences of a positive characteristic variant of the Bogomolov-Gieseker inequality, which we shall state in the surface case only (for simplicity of notation): Let \(X\) be a smooth projective surface over an algebraically closed field \(k\) with \(\text{char} (k) = p > 0\). Let \(E\) be a rank 2 vector bundle on \(X\). Assume that the \(i\)-th absolute Frobenius pull-back \(E^{(i)}\) of \(E\) is weakly \(\mu\)-semi-stable for \(0 \leq i<n\). If \(X\) is not uniruled then: \[ c_ 1 (E)^ 2 - 4c_ 2(E) \leq \sigma (X)/(p^ n - 1)^ 2 \] where \(\sigma (X) = \inf (K_ X \cdot D)/(D^ 2)\), the infimum being taken over \(D\) in the nef cone of \(X\). Besides specific vector bundles techniques, the author uses the numerical criterion for uniruledness of \textit{Y. Miyaoka} and \textit{S. Mori} [Ann. Math., II. Ser. 124, 65-69 (1986; Zbl 0606.14030)].
scientific article; zbMATH DE number 537261