## The Bogomolov-Miyaoka-Yau inequality for logarithmic surfaces in positive characteristic.(English)Zbl 1386.14160

The paper under review generalizes Bogomolov’s inequality for Higgs sheaves and the Bogomolov-Miyaoka-Yau inequality in positive characteristic. More precisely, the results are formulated in the following theorems.
Let $$k$$ be an algebraically closed field of characteristic $$p>0$$. Let $$X$$ be a smooth projective variety of dimension $$n\geq 2$$ over $$k$$. Let $$H$$ be an ample divisor on $$X$$. For a rank $$r$$ torsion-Free sheaf $$E$$ on $$X$$. let $$\Delta(E):=2rc_2(E)-(r-1)c_1(E)^2$$.
Theorem 1: Let $$D$$ be a normal crossing divisor on $$X$$. Assume that the pair $$(X,D)$$ admits a lifting to the Witt ring $$W_2(k)$$. Then for any slope $$H$$-semistable logarithmic Higgs sheaf $$(E,\theta: E\to E\otimes \Omega_X(\log\,D))$$ of rank $$r\leq p$$, we have $$\Delta(E)H^{n-2}\geq 0$$.
As an application of Theorem 1, when $$\dim(X)=2$$, we have
Theorem 2: Let $$D$$ be a normal crossing divisor on a smooth projective surface $$X$$ over $$k$$. Assume that $$(X,D)$$ admits a lifting to $$W_2(k)$$. If the Iitaka dimension satisfies $$\kappa(X,D)\geq 0$$ and $$(K_X+D)^2\geq 0$$, then the following inequalities hold: (1) if $$p=2$$, then $$(K_X+D)^2\leq 4c_2(\Omega_X(\log\, D))$$, and (2) if $$p\geq 3$$, then $$(K_X+D)^2\leq 3c_2(\Omega_X(\log\,D))$$.
N. I. Shepherd-Barron [Invent. Math. 106, No. 2, 243–262 (1991; Zbl 0769.14006)] proved that if a smooth projective surface $$X$$ is not of general type, then Bogomolov’s inequality holds for rank $$2$$ vector bundles unless $$X$$ is quasielliptic of Kodaira dimension $$1$$. The next result generalizes this to the higher rank case.
Theorem 3: Let $$X$$ be a smooth projective surface over $$k$$ with $$\kappa(X)\leq 1$$, and assume that on $$X$$ there exists a slope-semistable sheaf $$E$$ with $$\Delta(E)<0$$. Then $$\kappa(X)=1$$ and the Iitaka fibration of $$X$$ is quasielliptic (in particular, $$p=2$$ or $$p=3$$).
Using these results, examples of nonconnected curves on rational surfaces that cannot be lifted modulo $$p^2$$, are constructed.
Theorem 4: Given any field $$k$$, there exists a pair $$(X,D)$$ consisting of a smooth rational surface $$X$$ over $$k$$ and a smooth (but not connected) divisor $$D$$ over $$k$$ such that $$(X,D)$$ does not lift to $$W_2(k)$$.

### MSC:

 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14G17 Positive characteristic ground fields in algebraic geometry 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties

Zbl 0769.14006
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