Articles written in Proceedings – Mathematical Sciences
Volume 119 Issue 1 February 2009 pp 109-118
On some Frobenius Restriction Theorems for Semistable Sheaves
We prove a version of an effective Frobenius restriction theorem for semistable bundles in characteristic 𝑝. The main novelty is in restricting the bundle to the 𝑝-fold thickening of a hypersurface section. The base variety is $G/P$, an abelian variety or a smooth projective toric variety.
Volume 122 Issue 4 November 2012 pp 615-628
Frobenius Pull Backs of Vector Bundles in Higher Dimensions
We prove that for a smooth projective variety 𝑋 of arbitrary dimension and for a vector bundle 𝐸 over 𝑋, the Harder–Narasimhan filtration of a Frobenius pull back of 𝐸 is a refinement of the Frobenius pull back of the Harder–Narasimhan filtration of 𝐸, provided there is a lower bound on the characteristic 𝑝 (in terms of rank of 𝐸 and the slope of the destabilizing sheaf of the cotangent bundle of 𝑋). We also recall some examples, due to Raynaud and Monsky, to show that some lower bound on 𝑝 is necessary. We also give a bound on the instability degree of the Frobenius pull back of 𝐸 in terms of the instability degree of 𝐸 and well defined invariants of 𝑋.
Volume 133, 2023
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