Tohoku Mathematical Journal

On nef and semistable hermitian lattices, and their behaviour under tensor product

Yves André

Full-text: Open access

Abstract

We study the behaviour of semistability under tensor product in various settings: vector bundles, euclidean and hermitian lattices (alias Humbert forms or Arakelov bundles), multifiltered vector spaces.

One approach to show that semistable vector bundles in characteristic zero are preserved by tensor product is based on the notion of nef vector bundles. We revisit this approach and show how far it can be transferred to hermitian lattices. J.-B. Bost conjectured that semistable hermitian lattices are preserved by tensor product. Using properties of nef hermitian lattices, we establish an inequality in that direction.

We axiomatize our method in the general context of monoidal categories, and then give an elementary proof of the fact that semistable multifiltered vector spaces (which play a role in diophantine approximation) are preserved by tensor product.

Article information

Source
Tohoku Math. J. (2), Volume 63, Number 4 (2011), 629-649.

Dates
First available in Project Euclid: 6 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1325886284

Digital Object Identifier
doi:10.2748/tmj/1325886284

Mathematical Reviews number (MathSciNet)
MR2872959

Zentralblatt MATH identifier
1256.11030

Subjects
Primary: 11E39: Bilinear and Hermitian forms
Secondary: 14G25: Global ground fields 14H60: Vector bundles on curves and their moduli [See also 14D20, 14F05]

Citation

André, Yves. On nef and semistable hermitian lattices, and their behaviour under tensor product. Tohoku Math. J. (2) 63 (2011), no. 4, 629--649. doi:10.2748/tmj/1325886284. https://projecteuclid.org/euclid.tmj/1325886284


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