On the structure of instability in moduli theory

TL;DR

This paper develops a general theory of instability and stratifications for moduli problems in algebraic geometry, unifying classical concepts and extending to new contexts like Bridgeland stability.

Contribution

It introduces the notion of $ heta$-stratification for moduli problems, generalizing existing stratifications, and establishes conditions for their existence and applications.

Findings

Defined a $ heta$-stratification framework for moduli stacks.

Proved necessary and sufficient conditions for stratification existence.

Applied the theory to Bridgeland stability, beyond classical methods.

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 $\Theta$-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 $\Theta$-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.