Vũ Ngọc’s conjecture on focus-focus singular fibers with multiple pinched points

We classify, up to fiberwise symplectomorphisms, a saturated neighborhood of a singular fiber of an integrable system (which is proper onto its image and has connected fibers) containing \(k \geqslant 1\) focus-focus critical points. Our proof recovers the classification for \(k=1\) which was known prior to this paper. Our result shows that there is a one-to-one correspondence between such neighborhoods and k formal power series, up to a \((\mathbb {Z}_2 \times D_k)\)-action, where \(D_k\) is the kth dihedral group. The k formal power series determine the dynamical behavior of the Hamiltonian vector fields associated to the components of the momentum map on the symplectic manifold \((M,\omega )\) near the singular fiber containing the k focus-focus critical points. This proves a conjecture of San Vũ Ngọc from 2003.