Harder-Narasimhan polygons and Laws of Large Numbers

TL;DR

This paper applies probabilistic methods, specifically the Central Limit Theorem, to analyze the asymptotic behavior of Harder-Narasimhan polygons, expanding slope stability theory for filtered vector spaces with applications in algebraic geometry.

Contribution

It introduces a probabilistic approach to the study of Harder-Narasimhan polygons and extends slope stability theory for filtered vector spaces, building on recent techniques and classical methods.

Findings

Vertices of Harder-Narasimhan polygons exhibit asymptotic normal distribution.

Established a filtered vector space analogue of key stability results.

Connected lattice reduction methods to probabilistic analysis of geometric data.

Abstract

We build on the recent techniques of Codogni and Patakfalvi, from \cite{Codogni:Patakfalvi:2021}, which were used to establish theorems about semi-positivity of the Chow Mumford line bundles for families of $\K$-semistable Fano varieties. Here we apply the Central Limit Theorem to ascertain the asymptotic probabilistic nature of the vertices of the \emph{Harder and Narasimhan polygons}. As an application of our main result, we use it to establish a filtered vector space analogue of the main technical result of \cite{Codogni:Patakfalvi:2021}. In doing so, we expand upon the slope stability theory, for filtered vector spaces, that was initiated by Faltings and W\"{u}stholz \cite{Faltings:Wustholz}. One source of inspiration for our abstract study of \emph{Harder and Narasimhan data}, which is a concept that we define here, is the lattice reduction methods of Grayson \cite{Grayson:1984}. Another is the work of Faltings and W\"{u}stholz, \cite{Faltings:Wustholz}, and Evertse and Ferretti, \cite{Evertse:Ferretti:2013}, which is within the context of Diophantine approximation for projective varieties.