# Vu Ngoc's Conjecture on focus-focus singular fibers with multiple pinched points

## TL;DR

This paper classifies neighborhoods of focus-focus singular fibers with multiple pinched points in integrable systems, establishing a correspondence with formal power series and proving a conjecture by San Vu Ngoc from 2002.

## Contribution

It provides a complete classification of such neighborhoods using formal power series, confirming a long-standing conjecture in symplectic geometry.

## Key findings

- Classification up to fiberwise symplectomorphisms using formal power series
- Establishment of a one-to-one correspondence with formal power series
- Proof of San Vu Ngoc's conjecture from 2002

## Abstract

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 $k > 1$ focus-focus critical points. 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 $k$-th 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 Vu Ngoc from 2002.