Complex Hamiltonian systems on $\mathbb{C^2}$ with Hamiltonian function of low Laurent degree

N. N. Martynchuk

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2015), pp. 3-9

Résumé

We study complex Hamiltonian systems on $\mathbb C\times(\mathbb C\setminus\{0\})$ with standard symplectic structure $\omega_{\mathbb C}=dz\wedge dw$ and Hamiltonian function $f=a z^2+b/w+P_n(w)$, where $P_n(w)$ is a polynomial of degree $n$, the numbers $a,b\in\mathbb C$ and $ab\ne0$. For some natural classes of these $\mathbb C$-Hamiltonian systems we study an equivalence relation in the Hamiltonian sense and determine the topology of the corresponding quotient space. We also prove that for $\mathbb C$-Hamiltonian systems with Hamiltonian function $f=a z^2+b/w+P_n(w)$, where $a b\ne0,n\ge0$, the bifurcation complex is homeomorphic to a two-dimensional plane.