2021
DOI: 10.1007/s00209-021-02805-8
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Characteristic classes and stability conditions for projective Kleinian orbisurfaces

Abstract: We construct Bridgeland stability conditions on the derived category of smooth quasiprojective Deligne-Mumford surfaces whose coarse moduli spaces have ADE singularities. This unifies the construction for smooth surfaces and Bridgeland's work on Kleinian singularities. The construction hinges on an orbifold version of the Bogomolov-Gieseker inequality for slope semistable sheaves on the stack, and makes use of the Toën-Hirzebruch-Riemann-Roch theorem.

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Cited by 2 publications
(2 citation statements)
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References 24 publications
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“…The basic idea is the following. Using the results of Subsection 5.2 and the projection X → C we show that all ı-stable MT sheaves on X are given by Nironi δ-stable sheaves on [S/ı] × {t} for some t ∈ C. We then show that Nironi δ-stability on [S/ı] is the large volume limit of a certain Bridgeland stability condition on [S/ı] constructed by Lim and Rota [15]. We then apply the derived Fourier-Mukai correspondence to show that our moduli spaces are given by, up to a factor of C, moduli spaces of objects in the derived category of S which are stable with respect to the large volume limit of one of the stability conditions on K3 surfaces constructed by Bridgeland [6].…”
Section: Local Abelian and Nikulin Surfaces (Mt Theory)mentioning
confidence: 85%
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“…The basic idea is the following. Using the results of Subsection 5.2 and the projection X → C we show that all ı-stable MT sheaves on X are given by Nironi δ-stable sheaves on [S/ı] × {t} for some t ∈ C. We then show that Nironi δ-stability on [S/ı] is the large volume limit of a certain Bridgeland stability condition on [S/ı] constructed by Lim and Rota [15]. We then apply the derived Fourier-Mukai correspondence to show that our moduli spaces are given by, up to a factor of C, moduli spaces of objects in the derived category of S which are stable with respect to the large volume limit of one of the stability conditions on K3 surfaces constructed by Bridgeland [6].…”
Section: Local Abelian and Nikulin Surfaces (Mt Theory)mentioning
confidence: 85%
“…In [15], Lim and Rota construct Bridgeland stability conditions on orbifold surfaces with Kleinian orbifold points. For notational simplicity, they assume that their orbifold surface has a single orbifold point, but their method easily applies to orbifold surfaces with multiple Kleinian orbifold points such as [S/ı].…”
Section: Nironi Stabilitymentioning
confidence: 99%