Refined Harder-Narasimhan filtrations in moduli theory

Abstract

We define canonical refinements of Harder-Narasimhan filtrations and stratifications in moduli theory, generalising and relating work of Haiden-Katzarkov-Kontsevich-Pandit and Kirwan. More precisely, we define a canonical stratification on any noetherian algebraic stack $\mathcal X$ with affine diagonal that admits a good moduli space and is endowed with a norm on graded points. The strata live in a newly defined stack of sequential filtrations of $\mathcal X$. Therefore the stratification gives a canonical sequential filtration, the iterated balanced filtration, for each point of $\mathcal X$. Examples of applicability include moduli of principal bundles on a curve, moduli of objects at the heart of a Bridgeland stability condition and moduli of K-semistable Fano varieties. We conjecture that the iterated balanced filtration describes the asymptotics of the Kempf-Ness flow in Geometric Invariant Theory, as part of a larger project aiming to describe the asymptotics of natural flows in moduli theory. In the case of quotient stacks by diagonalisable algebraic groups, we give an explicit description of the iterated balanced filtration in terms of convex geometry.