On the symplectic geometry of $A_k$ singularities
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917709362495488 |
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| author | Martynchuk, Nikolay Ngoc, San Vũ |
| author_facet | Martynchuk, Nikolay Ngoc, San Vũ |
| contents | This paper presents a complete symplectic classification of $A_k$ Hamiltonians on $\mathbb R^2$, in the analytic and smooth categories. Precisely, consider the pair $(H, ω)$ consisting of a Hamiltonian and a symplectic structure on $\mathbb R^2$ such that $H$ has an $A_{k-1}$ singularity at the origin with $k\geq 2$. We classify such pairs near the origin, up to fiberwise symplectomorphisms, and up to $H$-preserving symplectomorphisms. The classification is obtained by bringing the pair $(H, ω)$ to a symplectic normal form $$\big(H = ξ^2 \pm x^k, \ ω= d (f d ξ)\big), \quad f = \sum_{i=1}^{k-1} x^i f_i(x^k),$$ modulo some relations which are explicitly given. We also show that the group of $H$-preserving symplectomorphisms of an $A_{k-1}$ singularity for $k$ odd consists of symplectomorphisms that can be included into a $C^\infty$-smooth (resp., real-analytic) $H$-preserving flow, whereas for $k$ even with $k \ge 4$ the same is true modulo the $\mathbb Z_2$-subgroup generated by the involution $Inv(x,ξ) = (-x,-ξ)$. The paper is concluded with a brief discussion of the conjecture that the symplectic invariants of $A_{k-1}$ singularities are spectrally determined. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_01814 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the symplectic geometry of $A_k$ singularities Martynchuk, Nikolay Ngoc, San Vũ Symplectic Geometry Differential Geometry Dynamical Systems 37J39, 37J35, 58J50, 70H06, 70H15 This paper presents a complete symplectic classification of $A_k$ Hamiltonians on $\mathbb R^2$, in the analytic and smooth categories. Precisely, consider the pair $(H, ω)$ consisting of a Hamiltonian and a symplectic structure on $\mathbb R^2$ such that $H$ has an $A_{k-1}$ singularity at the origin with $k\geq 2$. We classify such pairs near the origin, up to fiberwise symplectomorphisms, and up to $H$-preserving symplectomorphisms. The classification is obtained by bringing the pair $(H, ω)$ to a symplectic normal form $$\big(H = ξ^2 \pm x^k, \ ω= d (f d ξ)\big), \quad f = \sum_{i=1}^{k-1} x^i f_i(x^k),$$ modulo some relations which are explicitly given. We also show that the group of $H$-preserving symplectomorphisms of an $A_{k-1}$ singularity for $k$ odd consists of symplectomorphisms that can be included into a $C^\infty$-smooth (resp., real-analytic) $H$-preserving flow, whereas for $k$ even with $k \ge 4$ the same is true modulo the $\mathbb Z_2$-subgroup generated by the involution $Inv(x,ξ) = (-x,-ξ)$. The paper is concluded with a brief discussion of the conjecture that the symplectic invariants of $A_{k-1}$ singularities are spectrally determined. |
| title | On the symplectic geometry of $A_k$ singularities |
| topic | Symplectic Geometry Differential Geometry Dynamical Systems 37J39, 37J35, 58J50, 70H06, 70H15 |
| url | https://arxiv.org/abs/2305.01814 |