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arXiv · 1411.0627

On the structure of instability in moduli theory

Daniel Halpern-Leistner
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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.

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Daniel Halpern-Leistner. 2014-11-03. On the structure of instability in moduli theory. https://arxiv.org/abs/1411.0627

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