Harder-Narasimhan polygons and Laws of Large Numbers

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.