Noncommutative projective partial resolutions and quiver varieties
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912477640392704 |
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| author | Gammelgaard, Søren Gyenge, Ádám |
| author_facet | Gammelgaard, Søren Gyenge, Ádám |
| contents | Let $Γ\in \mathrm{SL}_2(\mathbb{C})$ be a finite subgroup. We introduce a class of projective noncommutative surfaces $\mathbb{P}^2_I$, indexed by a set of irreducible $Γ$-representations. Extending the action of $Γ$ from $\mathbb{C}^2$ to $\mathbb{P}^2$, we show that these surfaces generalise both $[\mathbb{P}^2/Γ]$ and $\mathbb{P}^2/Γ$. We prove that isomorphism classes of framed torsion-free sheaves on any $\mathbb{P}^2_I$ carry a canonical bijection to the closed points of appropriate Nakajima quiver varieties. In particular, we provide geometric interpretations for a class of Nakajima quiver varieties using noncommutative geometry. Our results partially generalise several previous results on such quiver varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00709 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Noncommutative projective partial resolutions and quiver varieties Gammelgaard, Søren Gyenge, Ádám Algebraic Geometry Rings and Algebras 14A22, 16G20, 14E16 Let $Γ\in \mathrm{SL}_2(\mathbb{C})$ be a finite subgroup. We introduce a class of projective noncommutative surfaces $\mathbb{P}^2_I$, indexed by a set of irreducible $Γ$-representations. Extending the action of $Γ$ from $\mathbb{C}^2$ to $\mathbb{P}^2$, we show that these surfaces generalise both $[\mathbb{P}^2/Γ]$ and $\mathbb{P}^2/Γ$. We prove that isomorphism classes of framed torsion-free sheaves on any $\mathbb{P}^2_I$ carry a canonical bijection to the closed points of appropriate Nakajima quiver varieties. In particular, we provide geometric interpretations for a class of Nakajima quiver varieties using noncommutative geometry. Our results partially generalise several previous results on such quiver varieties. |
| title | Noncommutative projective partial resolutions and quiver varieties |
| topic | Algebraic Geometry Rings and Algebras 14A22, 16G20, 14E16 |
| url | https://arxiv.org/abs/2406.00709 |