Towards a characterization of toric hyperkähler varieties among symplectic singularities
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915560684519424 |
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| author | Namikawa, Yoshinori |
| author_facet | Namikawa, Yoshinori |
| contents | Let $(X, ω)$ be a conical symplectic variety of dimension $2n$ which has a projective symplectic resolution. Assume that $X$ admits an effective Hamiltonian action of an $n$-dimensional algebraic torus $T^n$, compatible with the conical $\mathbf{C}^*$-action. A typical example of $X$ is a toric hyperkahler variety $Y(A,0)$.
In this article, we prove that this property characterizes $Y(A,0)$ with $A$ unimodular. More precisely, if $(X, ω)$ is such a conical symplectic variety, then there is a $T^n$-equivariant (complex analytic) isomorphism $φ: (X, ω) \to (Y(A,0), ω_{Y(A,0)})$ under which both moment maps are identified. Moreover $φ$ sends the center $0_X$ of $X$ to the center $0_{Y(A,0)}$ of $Y(A,0)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03012 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Towards a characterization of toric hyperkähler varieties among symplectic singularities Namikawa, Yoshinori Algebraic Geometry 14, 32 Let $(X, ω)$ be a conical symplectic variety of dimension $2n$ which has a projective symplectic resolution. Assume that $X$ admits an effective Hamiltonian action of an $n$-dimensional algebraic torus $T^n$, compatible with the conical $\mathbf{C}^*$-action. A typical example of $X$ is a toric hyperkahler variety $Y(A,0)$. In this article, we prove that this property characterizes $Y(A,0)$ with $A$ unimodular. More precisely, if $(X, ω)$ is such a conical symplectic variety, then there is a $T^n$-equivariant (complex analytic) isomorphism $φ: (X, ω) \to (Y(A,0), ω_{Y(A,0)})$ under which both moment maps are identified. Moreover $φ$ sends the center $0_X$ of $X$ to the center $0_{Y(A,0)}$ of $Y(A,0)$. |
| title | Towards a characterization of toric hyperkähler varieties among symplectic singularities |
| topic | Algebraic Geometry 14, 32 |
| url | https://arxiv.org/abs/2408.03012 |