On some Frobenius restriction theorems for semistable sheaves
Let \(X\) be a smooth projective variety defined over an algebraically closed field of characteristic \(p\). Let \(F_X:X\to X\) be the Frobenius morphism. We say that a sheaf is \textit{strongly semistable} if for every \(t\) the sheaf \((F_X^t)^*E\) is semistable (with respect to some fixed polarization). It is known that on some varieties, including homogeneous, abelian and toric varieties, a semistable sheaf is strongly semistable. But in general semistability and strong semistability are different notions and it is not known if a strongly semistable sheaf remains strongly semistable if restricted to a very general hypersurface of large degree. Related to this question, the authors prove the following result. Let \(E\) be a slope semistable sheaf on a homogeneous, abelian or toric variety. If the restrictions of all wedge powers of \(E\) remain semistable on a general hypersurface of degree \(d\) then \((F_X^t)^*E\) is semistable on a general hypersurface of degree \(dp^t\). Since the degree grows with \(t\) this does not provide restriction theorem for strong semistability. After this paper was written the reviewer proved an effective restriction theorem for strong semistability for a large class of varieties including the varieties considered by the authors [see \textit{A. Langer}, ``A note on restriction theorems for semistable sheaves, With an appendix: Semistability of the syzygy bundle on \(\mathbb{P}^n\) by V. B. Mehta'', preprint \url{arXiv:0905.4606}].
scientific article; zbMATH DE number 5601779