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

TL;DR

This paper investigates how semistability of hermitian lattices behaves under tensor product, showing that nef hermitian lattices are not preserved, but semistable multifiltered vector spaces are, with implications for Arakelov geometry and diophantine approximation.

Contribution

It extends the understanding of semistability preservation under tensor product from vector bundles to hermitian lattices, providing new inequalities and axiomatization in monoidal categories.

Findings

Nef hermitian lattices are not preserved by tensor product.

Semistable multifiltered vector spaces are preserved by tensor product.

An inequality improving previous results on hermitian lattices' semistability.

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 which improves on earlier results in that direction. On the other hand, we show that, in contrast to nef vector bundles, nef hermitian lattices are not preserved by tensor product. We axiomatize our method in the general context of monoidal categories, and give an elementary proof of the fact that semistable multifiltered vector spaces (which play a role in diophantine approximation) are preserved by tensor product.