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arXiv · 1803.00998

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

Álvaro Pelayo·Xiudi Tang
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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.

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Álvaro Pelayo, Xiudi Tang. 2018-03-02. Vu Ngoc's Conjecture on focus-focus singular fibers with multiple pinched points. https://arxiv.org/abs/1803.00998

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