A note on Bogomolov-Gieseker's inequality in positive characteristic
Let \(X\) be a \(d\)-dimensional nonsingular projective variety with an ample line bundle \(H\) and \(E\) a torsion-free sheaf of rank \(r\) on \(X\) that is so-called \(\mu\)-semistable with respect to \(H\). The Bogomolov-Gieseker inequality \((r-1)c_ 1(E)^ 2\leq 2rc_ 2(E)\) holds if \(\dim X=2\) and the characteristic is zero. This is proven in appendix B of the paper by establishing effective boundedness of families of torsionfree sheaves to obtain an effective estimation for the nonnegativity of \(c_ 2\). --- In appendix A, the author proves the inequality for positive characteristic if the \(n\)-th Frobenius pullback of \(E\) is \(\mu\)-semistable for all \(n\). The main part of the paper obtains, for arbitrary \(d\) and \(r\leq 3\), the inequality with on both sides the intersection product with \(H^{d-1}\) being taken. Appendix C is on the preservation of semistability under restriction to an ample curve.
scientific article; zbMATH DE number 58552
scientific article