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Inequality of Bogomolov-Gieseker type on arithmetic surfaces. (English) Zbl 0854.14012

On an \(n\)-dimensional compact Kähler manifold with a Kähler form \(\Phi\), for any \(\Phi\)-semistable torsion-free sheaf \(E\) of rank \(r\), the geometric Bogomolov-Gieseker inequality is \[ \int_M \left (c_2(E) - {r - 1 \over 2r} c_1 (E)^2 \right) \wedge \Phi^{n - 2} \geq 0; \] in particular, for a semistable vector bundle on an algebraic surface, this inequality gives \(c_1 (E)^2 \leq (\{2r\}/(r - 1)) c_2 (E)\). Miyaoka provided an arithmetic analogue of the above inequality for stable vector bundles on an arithmetic surface. The paper under review obtains this inequality for any semistable vector bundle on an arithmetic surface; more precisely, the author proves the following main theorem: Let \(f : X \to \text{Spec} ({\mathcal O}_K)\) be an arithmetic surface and \((E,h)\) a Hermitian vector bundle on \(X\). If \(E_{\overline \mathbb{Q}}\) is semistable on the geometric generic fibre \(X_{\overline \mathbb{Q}}\) of \(f\), then \[ \widehat c_2 (E,h) - {r - 1 \over 2r} \widehat c_1 (E,h)^2 \geq 0 \] where \(r = \text{rk} (E)\) and \(\widehat c_1 (E,h)\), \(\widehat c_2 (E,h)\) are the arithmetic Chern classes introduced by Gillet and Soule.
After a clever calculation of the secondary Bott-Chern characteristic, the author reduces the theorem to the stable case; then, using Gillet-Soulé’s arithmetic Riemann-Roch theorem, it is enough to prove the main theorem if one has a good estimation for the \(L^2\)-degree of a Hermitian module and analytic torsions, which requires that the vector bundle \(E\) have a Hermitian-Einstein metric. The method is different from Miyaoka’s. [For Miyaoka’s proof, see C. Soulé’s paper: Invent. Math. 116, No. 1-3, 577-599 (1994; Zbl 0834.14013).] The paper under review also contains some other interesting results.

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14F45 Topological properties in algebraic geometry
57R20 Characteristic classes and numbers in differential topology

Citations:

Zbl 0834.14013
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References:

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