It is well-known that [Miyaoka][2] and [Yau][1]-type inequalities do not hold in positive characteristic. In "[a note on Bogomolov-Gieseker’s inequality in positive characteristic][3]", however, we can find the following
> <b>Theorem 1.</b> Assume that $p = \mathrm{char}(k) > 0$. Let $X$ be a $d$-dimensional nonsingular
projective $k$-variety with an ample line bundle $H$. Let $E$ be a $p$-semistable (with respect to $H$) vector bundle of rank $r$ on $X$. Assume $d \ge 2$. Then we have
$$(r-1)(c_1(E)^2 \cdot H^{d-2}) \le 2r(c_2(E)\cdot H^{d-2})$$
if <b>(1)</b> $r < 3$ or <b>(2)</b> $d = 2$.

My question is, are there similar, further results that relate numerical invariants of a nonsingular projective variety in positive characteristic? In particular, it would help me a great deal if there was one involving also the characteristic $p=\mathrm{char}(k)$ itself.

  [1]: https://www.pnas.org/doi/pdf/10.1073/pnas.74.5.1798
  [2]: http://www.springerlink.com/content/q3v55834r8615281/fulltext.pdf
  [3]: https://projecteuclid.org/journals/duke-mathematical-journal/volume-64/issue-2/A-note-on-Bogomolov-Giesekers-inequality-in-positive-characteristic/10.1215/S0012-7094-91-06418-5.short