On the structure of instability in moduli theory

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.

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

References in corpus (3)

• Alternate Compactifications of Moduli Spaces of Curves
• Quotients of unstable subvarieties and moduli spaces of sheaves of fixed Harder-Narasimhan type
• Schematic HN stratification for families of principal bundles and lambda modules

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