Title: Smooth invariants of focus-focus singularities and obstructions to 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.