On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917982683267072 |
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| author | Hu, Mengwei |
| author_facet | Hu, Mengwei |
| contents | Kleinian singularities are quotients of $\mathbb{C}^2$ by finite subgroups of $\mathrm{SL}_2(\mathbb{C})$. They are in bijection with the simply-laced Dynkin diagrams via the McKay correspondence. Anti-Poisson involutions and their fixed point loci appear naturally when we want to classify irreducible Harish-Chandra modules over Kleinian singularities. There are three goals of this paper. The first is to classify anti-Poisson involutions of Kleinian singularities up to conjugation by graded Poisson automorphisms. The second is to describe the scheme-theoretic fixed point loci of Kleinian singularities under anti-Poisson involutions. The last and the main goal is to describe the scheme-theoretic preimages of the fixed point loci under minimal resolutions of Kleinian singularities, which are singular Lagrangian subvarieties in the minimal resolutions whose irreducible components are $\mathbb{P}^1$'s and $\mathbb{A}^1$'s. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_08717 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities Hu, Mengwei Representation Theory Algebraic Geometry 16G20, 14H20 Kleinian singularities are quotients of $\mathbb{C}^2$ by finite subgroups of $\mathrm{SL}_2(\mathbb{C})$. They are in bijection with the simply-laced Dynkin diagrams via the McKay correspondence. Anti-Poisson involutions and their fixed point loci appear naturally when we want to classify irreducible Harish-Chandra modules over Kleinian singularities. There are three goals of this paper. The first is to classify anti-Poisson involutions of Kleinian singularities up to conjugation by graded Poisson automorphisms. The second is to describe the scheme-theoretic fixed point loci of Kleinian singularities under anti-Poisson involutions. The last and the main goal is to describe the scheme-theoretic preimages of the fixed point loci under minimal resolutions of Kleinian singularities, which are singular Lagrangian subvarieties in the minimal resolutions whose irreducible components are $\mathbb{P}^1$'s and $\mathbb{A}^1$'s. |
| title | On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities |
| topic | Representation Theory Algebraic Geometry 16G20, 14H20 |
| url | https://arxiv.org/abs/2504.08717 |