On the symplectic geometry of \(A_k\) singularities

• Published in: arXiv.org 2024-07
• Main Authors: Martynchuk, Nikolay, San, Vũ Ngoc
• Format: Article
• Language: English
• Subjects: Canonical forms; Classification; Singularities; Subgroups
• ISSN: 2331-8422

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 \((H, \omega)\) 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 \((H, \omega)\) to a symplectic normal form $$\big(H = \xi^2 \pm x^k, \ \omega = d (f d \xi)\big), \quad f = \sum_{i=1}^{k-1} x^i f_i(x^k),$$ 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 \(Inv(x,\xi) = (-x,-\xi)\). The paper is concluded with a brief discussion of the conjecture that the symplectic invariants of \(A_{k-1}\) singularities are spectrally determined.