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

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.