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**Arithmetic Bogomolov-Gieseker’s inequality.**
*(English)*
Zbl 0854.14013

Let \(f:X\to\text{Spec} \mathbb{Z}\) be an arithmetic variety of absolute dimension \(d\). A Hermitian line bundle \((H,k)\) on \(X\) is called arithmetically ample if \(H\) is \(f\)-ample, the first Chern form \(c_1(H_\mathbb{C},k)\) is a Kähler form on \(X(\mathbb{C})\), and for every irreducible horizontal subvariety \(Y\) of \(X\) the height \(\widehat c_1((H,k)|_Y)^{\dim Y}\) is positive. Let \((H,k)\) be an arithmetically ample Hermitian line bundle on \(X\) and assume that \(d\geq 2\). We consider a Hermitian vector bundle \((E,h)\) on \(X\) of rank \(r\) such that the induced bundle \(E_\mathbb{C}\) is semistable with respect to \(H_\mathbb{C}\) on each connected component of the complex manifold \(X(\mathbb{C})\). The arithmetic Bogomolov-Gieseker inequality states that in this situation
\[
\left\{\widehat c_2(E,h)-{(r-1)\over 2r}\widehat c_1(E,h)^2\right\}\cdot\widehat c_1(H,k)^{d-2}\geq 0.
\]
Moreover, if equality holds, then \(E_\mathbb{C}\) is projectively flat and \(h\) is a weakly Einstein Hermitian metric. The proof of the inequality proceeds (as in the geometric case) by induction on the dimension of \(X\). The case of arithmetic surfaces \((d=2)\) is treated in the author’s paper [Duke Math. J. 74, No. 3, 713-761 (1994; see the preceding review)]. In order to reduce the general inequality to this case, one uses a section \(s\in H^0(X,H)\) which satisfies the following:

(i) \(\text{div}(s_\mathbb{C})\) is smooth. (ii) \(E_\mathbb{C}|_{\text{div}}(s_\mathbb{C})\) is semistable. (iii) \(|s_\mathbb{C}|_{\sup}< 1\).

Using results of S. Zhang about sections of small norm of arithmetically ample Hermitian line bundles, one can prove an arithmetic version of Bertini’s theorem which assures the existence of a section \(s\) satisfying (i) and (iii). In order to obtain a section satisfying (i)–(iii), one uses Bogomolov’s restriction theorem. This result was proved by Bogomolov in the surface case and is generalized in the paper to higher-dimensional varieties. It states the following. Let \(X\) be a smooth projective variety of dimension \(d\geq 2\) over an algebraically closed field of characteristic zero, and let \(H\) be an ample divisor on \(X\). Let \(E\) be a semistable torsion-free sheaf on \(X\) with respect to \(H\). Then there are an effectively determined integer \(m_0\) and closed points \(x_1, \dots, x_s\) of \(X\) such that, for all \(m \geq m_0\), if \(C \in |mH |\) is normal and \(C \cap \{x_1, \dots, x_s\}=\emptyset\), then \(E |_C\) is semistable with respect to \(H |_C\).

(i) \(\text{div}(s_\mathbb{C})\) is smooth. (ii) \(E_\mathbb{C}|_{\text{div}}(s_\mathbb{C})\) is semistable. (iii) \(|s_\mathbb{C}|_{\sup}< 1\).

Using results of S. Zhang about sections of small norm of arithmetically ample Hermitian line bundles, one can prove an arithmetic version of Bertini’s theorem which assures the existence of a section \(s\) satisfying (i) and (iii). In order to obtain a section satisfying (i)–(iii), one uses Bogomolov’s restriction theorem. This result was proved by Bogomolov in the surface case and is generalized in the paper to higher-dimensional varieties. It states the following. Let \(X\) be a smooth projective variety of dimension \(d\geq 2\) over an algebraically closed field of characteristic zero, and let \(H\) be an ample divisor on \(X\). Let \(E\) be a semistable torsion-free sheaf on \(X\) with respect to \(H\). Then there are an effectively determined integer \(m_0\) and closed points \(x_1, \dots, x_s\) of \(X\) such that, for all \(m \geq m_0\), if \(C \in |mH |\) is normal and \(C \cap \{x_1, \dots, x_s\}=\emptyset\), then \(E |_C\) is semistable with respect to \(H |_C\).

Reviewer: K.Künnemann (MR 96i:14022)

### MSC:

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |

14F45 | Topological properties in algebraic geometry |

57R20 | Characteristic classes and numbers in differential topology |