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 131, 2021
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