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

Yves André

doi:10.2748/tmj/1325886284

2011 On nef and semistable hermitian lattices, and their behaviour under tensor product

Yves André

Tohoku Math. J. (2) 63(4): 629-649 (2011). DOI: 10.2748/tmj/1325886284

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.

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Yves André "On nef and semistable hermitian lattices, and their behaviour under tensor product," Tohoku Mathematical Journal 63(4), 629-649, (2011). https://doi.org/10.2748/tmj/1325886284

Published: 2011

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