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

@article{Pelayo2018VuNC, title={Vu Ngoc's Conjecture on focus-focus singular fibers with multiple pinched points}, author={'Alvaro Pelayo and Xiudi Tang}, journal={arXiv: Symplectic Geometry}, year={2018} }

We classify, up to symplectomorphisms, a neighborhood of a singular fiber of an integrable system (which is proper and has connected fibers) containing $k > 1$ non-degenerate 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. This proves a conjecture of San Vu Ngoc from 2002.

## 9 Citations

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