On the structure of instability in moduli theory

2014/11/03 by Daniel Halpern-Leistner, Halpern-Leistner, Daniel · 11 citations
Mathematics · Physics and Astronomy · #14D20 #14D23 #14L24 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Nonlinear Waves and Solitons #math.AG #msc:14D20 #msc:14D23 #msc:14L24

paper · pdf · doi:10.48550/arxiv.1411.0627

Final version, 173 pages, default margins. 1 figure. Minor corrections and improvements throughout the paper. Major additions since the last version: i) the notion of monotonicity for numerical invariants, 2) discussion of existence of moduli spaces

openalex publication_date 2014/11/03 · arxiv created 2022/02/04 · arxiv updated 2022/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We formulate a theory of instability and Harder-Narasimhan filtrations for an arbitrary moduli problem in algebraic geometry. We introduce the notion of a Θ-stratification of a moduli problem, which generalizes the Kempf-Ness stratification in GIT as well as the Harder-Narasimhan stratification of the moduli of coherent sheaves on a projective scheme. Our main theorems establish necessary and sufficient conditions for the existence of these stratifications. We define a structure on an algebraic stack called a numerical invariant, and we show that in many situations a numerical invariant defines a Θ-stratification on the stack, assuming a certain "HN boundedness" condition holds. We also discuss criteria under which the semistable locus has a moduli space. We apply our methods to an example that lies beyond the reach of classical methods: the stratification of the stack of objects in the heart of a Bridgeland stability condition.

Citations

• Schematic HN stratification for families of principal bundles and lambda modules

Cited by

• Motivic DT/PT Correspondence
• Isomorphisms between moduli stacks of vector bundles with fixed determinant
• Harder-Narasimhan filtrations of decorated vector bundles
• A categorical perspective on non-abelian localization
• The BPS decomposition theorem
• Stability and disconnected groups
• Moduli of non-pure sheaves and contractions of Hilbert schemes
• Remarks on Theta-stratifications and derived categories
• Modules and generalizations of Joyce vertex algebras
• Moduli spaces of twisted equivariant G-bundles over a curve
• Moduli spaces for Θ-strata and non-reductive quotients

Related

• A guide to moduli theory beyond GIT
• Refined Harder-Narasimhan filtrations in moduli theory
• Moduli Spaces of Unstable Objects: Sheaves of Harder-Narasimhan Length 2
• Theta-stratifications, Theta-reductive stacks, and applications
• Quotients by Parabolic Groups and Moduli Spaces of Unstable Objects