# Smooth invariants of focus-focus singularities and obstructions to product decomposition

## TL;DR

This paper investigates the smooth classification of focus-focus singularities in symplectic 4-manifolds, revealing new invariants, moduli space structure, and providing counterexamples to a conjecture on product decomposition.

## Contribution

It introduces smooth invariants for focus-focus singularities, describes their moduli space algebraically, and disproves Zung's conjecture on singularity decomposition.

## Key findings

- Existence of homeomorphic but not diffeomorphic focus-focus singularities.
- Moduli space of focus-focus singularities is one-dimensional for double pinched tori.
- Counterexamples to Zung's conjecture on singularity product decomposition.

## Abstract

We study focus-focus singularities (also known as nodal singularities, or pinched tori) of Lagrangian fibrations on symplectic $4$-manifolds. We show that, in contrast to elliptic and hyperbolic singularities, there exist homeomorphic focus-focus singularities which are not diffeomorphic. Furthermore, we obtain an algebraic description of the moduli space of focus-focus singularities up to smooth equivalence, and show that for double pinched tori this space is one-dimensional. Finally, we apply our construction to disprove Zung's conjecture which says that any non-degenerate singularity can be smoothly decomposed into an almost direct product of standard singularities.