A Remark on the Topology of the Regular Loci of Some Complexified Hamiltonian Systems

Abstract

We study the topology of the regular loci of two complexified Hamiltonian integrable systems using the Zariski-van Kampen method. In particular, we show that the fundamental group of the regular locus for the complexified planar Kepler problem is the free Abelian group $\mathbb{Z}\oplus \mathbb{Z}$, whereas that for the complexified spherical pendulum is $\mathbb{Z}$. These results further provide a description of the complex Hamiltonian monodromy group associated to these systems.