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**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 \(W_2(k)\).

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 \(W_2(k)\).

Reviewer: Noriko Yui (Kingston)

### 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 |