BOLSINOV, A.V. and IZOSIMOV, A., 2018. Smooth invariants of focus-focus singularities and obstructions to product decomposition. Journal of Symplectic Geometry, In Press.
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.
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