Vertices of the Harder and Narasimhan polygons and the laws of large numbers

Abstract We build on the recent techniques of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894), which were used to establish theorems about semi-positivity Chow Mumford line bundles for families $\mathrm {K}$ -semistable Fano varieties. Here, we apply Central Limit Theorem ascertain asymptotic probabilistic nature vertices Harder Narasimhan polygons . As an application our main result, use it a filtered vector space analogue technical result 811–894). In doing so, expand upon slope stability theory, spaces, that was initiated by Faltings Wüstholz (1994, 116, 109–138). One source inspiration abstract study data , is concept define here, lattice reduction methods Grayson (1984, Commentarii Mathematici Helvetic 59, 600–634). Another work 109–138), Evertse Ferretti (2013, Annals Mathematics 177, 513–590), within context Diophantine approximation projective