Bogomolov instability of higher rank sheaves on surfaces in characteristic \(p\) (Q1891213)

Let \(X\) be a smooth projective surface over an algebraically closed field of characteristic \(p > 0\), and let \({\mathcal E}\) be a locally free sheaf of rank \(r\) on \(X\) with \(\delta ({\mathcal E}) = (r - 1) c^ 2_ 1 ({\mathcal E}) - 2rc_ 2 ({\mathcal E}) > 0\). We prove that then some pull-back of \({\mathcal E}\) by the Frobenius morphism \(F : X \to X\) is Bogomolov unstable, i.e., there exists \(n \in \mathbb{N}\) such that \(F^{n*} {\mathcal E}\) has a destabilising subsheaf \({\mathcal E}_ 1\) which satisfies the following two conditions: \[ {1 \over p^{2n}} \left( c_ 1 ({\mathcal E}_ 1) - {\text{rk}{\mathcal E}_ 1 \over r} c_ 1 (F^{n*} {\mathcal E}) \right)^ 2 \geq {\delta ({\mathcal E}) \over 2r}, \quad \text{and} \quad \left( c_ 1 ({\mathcal E}_ 1) - {\text{rk} {\mathcal E}_ 1 \over r} c_ 1 (F^{n*} {\mathcal E})\right).\;H > 0 \] for any ample divisor \(H\) on \(X\).