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It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can find the following

Theorem 1. 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 (1) $r < 3$ or (2) $d = 2$.

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.

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Did you have a look at both of Shepherd-Barron's papers in Invent. Math. 106 (1991)? They're about surfaces, but go in the direction you're looking for... – Christian Liedtke Nov 10 '11 at 21:19
I had a look, this looks promising, thanks a lot! In particular, Corollary 15 from "Geography for surfaces of general type in positive characteristic" is nice. Of course, I would be very interested in any results for arbitrary dimension (and maybe even without bounds on the rank $r$), but I suppose that's a pipe dream. – Jesko Hüttenhain Nov 11 '11 at 17:00

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