On the symplectic geometry of $A_k$ singularities

Abstract

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 $k\ge 2$. 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 $$(H={\xi }^2\pm x^k,\omega =d(fd\xi )),f=\sum _{i=1}^{k-1}{x}^i{f}_i(x^k),$$ 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.