Reduction of the Semistability Condition for Tensors

TL;DR

This paper investigates the stability conditions of tensors, a special class of vector bundles with morphisms, and shows that overly long destabilizing filtrations can be shortened, simplifying stability analysis.

Contribution

It proves that long destabilizing filtrations of tensors can be replaced by shorter ones, reducing complexity in stability criteria and analyzing related combinatorial problems.

Findings

Shorter destabilizing filtrations exist for tensors.

A combinatorial framework for tensor filtrations is developed.

Semistable tensors on the projective line are characterized.

Abstract

In this article we study a special class of vector bundles, called tensors. A tensor consists of a vector bundle $E$ over a smooth irreducible projective variety and a morphism of vector bundles $\varphi$. As for classical vector bundles, there exists a notion of stability for these objects given in terms of filtrations of the vector bundle $E$. The aim of the present paper is to prove that if a destabilizing filtration is "too" long then there exists a shorter subfiltration which destabilizes as well. Moreover, we describe some related combinatorial problems, which arise from the description of a tensor $(E,\varphi)$ or, more precisely, a filtration of $E$ as a $a$-dimensional matrix. Eventually, as example we study semistable tensors on the projective line.