On the structure of instability in moduli theory
By Daniel Halpern-Leistner. 2014
89 citations recorded by the source corpus.
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 for coherent sheaves on a projective scheme, and establish necessary and sufficient conditions for the existence of these stratifications. For a certain class of algebraic stacks, which we call $\Theta$-reductive, the existence of $\Theta$-stratifications can be reduced to solving a relatively straightforward "boundedness" problem. 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.
Subfields: Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/bbf46adec058324df467afe1410a6b205ae025be
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Citations in the corpus
- Existence of moduli spaces for algebraic stacks (2018, methodology): “will sometimes say that a closed substack S⊂X is a Θ-stratum, if there exist a union of connected components S′ ⊂Map S (Θ,X) such that ev1: S′ →S⊂Xis an isomorphism. Our notation differs slightly from [Hal14], which denotes the stack Map S (Θ,X) by Filt(X) to promote the analogy of maps Θk →Xas filter”
- Reductivity of the automorphism group of K-polystable Fano varieties (2019, background): “k Xover k is Θ-reductive if for any DVR R and any diagram ΘR \0 // X ΘR << (3) there exists a unique dotted arrow filling in the diagram. Remark 2.8. This definition was introduced in [HL18]. As with S-completeness, if Xis an Artin stack with affine diagonal, then any lift is automatically un”