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This chapter is a continuation of the study undertaken in Arbarello and Saccà [Adv. Math., 329 649–703, (2018)]. We examine the local structure of Bridgeland moduli spaces \(M_\sigma (v,\mathcal {D})\) , where the relevant triangulated category \(\mathcal {D}\) is either the bounded derived category \(\mathcal {D}=\mathcal {D}^b(X)\) of a K3 surface X or the Kuznetsov component \(\mathcal {D}=\operatorname {Ku}(Y)\subset \mathcal {D} ^b(Y)\) of a smooth cubic fourfold \(Y\subset \mathbb {P}^5\) . For these moduli spaces, building on Bandiera, Manetti, and Meazzini [Moscow Mathematical Journal. 22, 239–263, (2022); Compositio Mathematica, 157 (2), 215–235, (2021)] we give a direct proof of formality, and, using their local isomorphism with quiver varieties, we establish their normality and their irreducibility, as long as \(\sigma \) does not lie on a totally semistable wall. We then connect the variation of GIT quotients for quiver varieties with the changing of stability conditions on moduli spaces.

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Singularities of Bridgeland Moduli Spaces for K3 Categories: An Update

  • 章节

  • 发表日期：

    2025年

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  • Enrico Arbarello,
  • Giulia Saccà

摘要

This chapter is a continuation of the study undertaken in Arbarello and Saccà [Adv. Math., 329 649–703, (2018)]. We examine the local structure of Bridgeland moduli spaces \(M_\sigma (v,\mathcal {D})\) , where the relevant triangulated category \(\mathcal {D}\) is either the bounded derived category \(\mathcal {D}=\mathcal {D}^b(X)\) of a K3 surface X or the Kuznetsov component \(\mathcal {D}=\operatorname {Ku}(Y)\subset \mathcal {D} ^b(Y)\) of a smooth cubic fourfold \(Y\subset \mathbb {P}^5\) . For these moduli spaces, building on Bandiera, Manetti, and Meazzini [Moscow Mathematical Journal. 22, 239–263, (2022); Compositio Mathematica, 157 (2), 215–235, (2021)] we give a direct proof of formality, and, using their local isomorphism with quiver varieties, we establish their normality and their irreducibility, as long as \(\sigma \) does not lie on a totally semistable wall. We then connect the variation of GIT quotients for quiver varieties with the changing of stability conditions on moduli spaces.