Refined Harder–Narasimhan filtrations in moduli theory

Thesis

We define canonical refinements of Harder–Narasimhan filtrations and stratifications in moduli theory. More precisely, we construct a canonical stratification for every noetherian algebraic stack $\mathcal{X}$ that has affine diagonal, 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 $\mathrm{Filt}_{\mathbb{Q}^\infty}(\mathcal{X})$ of $\mathcal{X}$. Therefore, the stratification gives a canonical sequential filtration, the iterated balanced filtration, for each point of $\mathcal{X}$. In the presence of suitable $\Theta$-stratifications, the iterated balanced filtration provides canonical refinements of Harder–Narasimhan filtrations. Our construction extends Kirwan's refined stratifications beyond the Geometric Invariant Theory case, and the iterated Haiden–Katzarkov–Kontsevich–Pandit (HKKP) filtration beyond the case of linear moduli problems, as well as giving a precise link between the two theories.

We introduce the machinery of chains of stacks as a tool to compute the iterated balanced filtration in combinatorial frameworks. We use it to give, in the case of quotient stacks by diagonalisable algebraic groups, an explicit description of the iterated balanced filtration in terms of convex geometry.

When the stack $\mathcal{X}$ parametrises objects in an abelian category, we show that the iterated balanced filtration agrees with the iterated HKKP filtration for modular lattices. A key ingredient in the proof is a new characterisation of the HKKP filtration for lattices as the minimiser of a certain norm function on the set of paracomplemented filtrations.

Examples where our theory can be applied include moduli of principal bundles on a curve, moduli of objects in 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. This would extend a theorem of Haiden, Katzarkov, Kontsevich and Pandit for quiver representations. This is part of a larger project aiming to describe the asymptotics of natural flows in moduli theory.

Authors: Ibáñez Núñez, A

• DOI: 10.5287/ora-eq09jjv80
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: geometric invariant theory; moduli spaces; algebraic geometry
• Subjects: Geometry