Concerning the semistability of tensor products in Arakelov geometry. (English) Zbl 1286.14040

Bost, Jean-Benoît; Chen, Huayi

J. Math. Pures Appl. (9) 99, No. 4, 436-488 (2013).

Let \(k\) be a field of characteristic \(0\), and let \(E\) and \(F\) be two vector bundles on a smooth projective integral curve \(C\) over \(k\), then it is well known that the semistabilities of \(E\) and \(F\) imply the semistability of \(E\otimes F\). The aim of the article under review is to investigate the arithmetic analogue of this result in the context of Arakelov geometry. Precisely, for any two Hermitian vector bundles \(\overline{E}\) and \(\overline{F}\) over an arithmetic curve \(\text{Spec}(\mathcal{O}_K)\) where \(\mathcal{O}_K\) is the ring of integers in a number field \(K\), the authors showed that the maximal slopes of \(\overline{E}\), \(\overline{F}\) and \(\overline{E}\otimes\overline{F}\) satisfy the inequality \[ \widehat{\mu}_{\text{max}}(\overline{E}\otimes\overline{F})\leq\widehat{\mu}_{\text{max}}(\overline{E})+\widehat{\mu}_{\text{max}}(\overline{F}) +\frac{1}{2}\text{min}(\log(\text{rk}E),\log(\text{rk}F)). \] They also proved that \(\widehat{\mu}(\overline{V})\leq\widehat{\mu}_{\text{max}}(\overline{E})+\widehat{\mu}_{\text{max}}(\overline{F})\) if \(V\) is an \(\mathcal{O}_K\)-submodule of \(E\otimes F\) of rank \(\leq 4\). This fact will imply that, if \(\overline{E}\) and \(\overline{F}\) are semistable and if \(\text{rk}E.\text{rk}F\leq 9\), then \(\overline{E}\otimes\overline{F}\) also is semistable.

Reviewer: Shun Tang (Beijing)

MSC:

14G40

Arithmetic varieties and schemes; Arakelov theory; heights

Keywords:

Hermitian vector bundle; tensor product; semi-stability

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