On nef and semistable Hermitian lattices, and their behaviour under tensor product. (English) Zbl 1256.11030

André, Yves

Tohoku Math. J. (2) 63, No. 4, Centen. Iss., 629-649 (2011).

Let \(\overline{E}\) be a \(\mathbb{Z}\)-lattice of full rank on a finite-dimensional euclidean vector space. The degree of \(\overline{E}\) is defined as \(\widehat{\deg}(\overline{E})=-\log (\text{vol}(\overline{E}))\), and the slope of \(\overline{E}\) as \(\mu(\overline{E})= \widehat{\deg}(\overline{E})/\text{rk}(\overline{E})\). The supremum of all slopes of all sublattices of any rank of \(\overline{E}\) is denoted by \(\mu_{\max}(\overline{E})\), and \(\overline{E}\) is called semistable if \(\mu(\overline{E})=\mu_{\max}(\overline{E})\). For example, integral unimodular lattices are semistable. For any pair \(\overline{E}_1, \overline{E}_2\) of nonzero euclidean lattices one can show that \(\mu(\overline{E}_1\otimes\overline{E}_2)= \mu(\overline{E}_1)+\mu(\overline{E}_2)\), and J.-B. Bost conjectured in the 1990s that this relation also holds for \(\mu_{\max}\). It follows from the result for \(\mu\) that \(\mu_{\max}(\overline{E}_1\otimes\overline{E}_2)\geq \mu_{\max}(\overline{E}_1)+\mu_{\max}(\overline{E}_2)\). The best known upper bound for \(\mu_{\max}\) of the tensor product is given by \(\mu_{\max}(\overline{E}_1\otimes\overline{E}_2)\leq \mu_{\max}(\overline{E}_1)+\mu_{\max}(\overline{E}_2)+ \frac{1}{2}\log(\text{rk}(\overline{E}_1\otimes\overline{E}_2))\). The author gives a new and elementary proof of this inequality.
If one replaces lattices by vector bundles \(E\) over a smooth projective curve \(S\) over a field of characteristic \(0\), one defines in complete analogy the slope of \(E\) as \(\mu (E)=\deg(E)/\text{rk}(E)\), and similarly \(\mu_{\max}\) by considering nonzero subbundles. Semistability can also be expressed in the same way as above. In this context, it is known for some time that the analogue of Bost’s conjecture holds and it was first shown by M. S. Narasimhan and C. S. Seshadri [Ann. Math. (2) 82, 540–567 (1965; Zbl 0171.04803)]. Another proof relies on the fact that a degree zero vector bundle is semistable if and only if it is a so-called nef (numerical efficiency) vector bundle, i.e. if its pull-back along any finite covering of \(S\) has no quotient line bundle of negative degree. The author gives a new simple version of this proof which also works for strongly semistable vector bundles in positive characteristic.
He then considers hermitian lattices \(\overline{E}\) over \(\overline{S}=\text{spec}(\mathfrak{o}_K)\cup V_{\infty}\) where \(K\) is a number field with ring of integers \(\mathfrak{o}_K\) and set of archimedean places \(V_{\infty}\). Defining the volume by taking appropriate products over \(V_{\infty}\), one defines analogous notions of \(\widehat{\deg}\), \(\mu\), \(\mu_{\max}\). Improving on earlier results by Chen and by Bost and Künnemann, the author shows that for hermitian lattices \(\overline{E}_1, \overline{E}_2\) over \(\overline{S}\), one has \[ \mu_{\max}(\overline{E}_1\otimes\overline{E}_2)\leq \mu_{\max}(\overline{E}_1)+\mu_{\max}(\overline{E}_2)+ \frac{[K:\mathbb{Q}]}{2}\log(\text{rk}(\overline{E}_1\otimes\overline{E}_2)) \]
The author develops a categorical setting in which all these cases with their definitions are just special cases, and considers another special case, namely what he calls generalized vector bundles with appropriate notions of slope and semistability of which the so-called multifiltered vector spaces are a subcase. If \(\overline{M}_1, \overline{M}_2\) are multifiltered spaces, then again one has the relation \(\mu_{\max}(\overline{M}_1\otimes\overline{M}_2)= \mu_{\max}(\overline{M}_1)+\mu_{\max}(\overline{M}_2)\). There are several proofs in the literature. The author provides a completely new and elementary proof inspired by his own proof of the result on euclidean lattices mentioned above.

Reviewer: Detlev Hoffmann (Dortmund)

MSC:

11E39	Bilinear and Hermitian forms
14G25	Global ground fields in algebraic geometry
14H60	Vector bundles on curves and their moduli

Keywords:

euclidean lattice; hermitian lattice; vector bundle; semistable vector bundle; nef vector bundle; slope; tensor product; multifiltered vector space

Citations:

Zbl 0171.04803

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