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

Alexey Bolsinov, Anton Izosimov

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1613-1648
Number of pages36
JournalJournal of Symplectic Geometry
Issue number6
StatePublished - 2019

ASJC Scopus subject areas

  • Geometry and Topology


Dive into the research topics of 'Smooth invariants of focus-focus singularities and obstructions to product decomposition'. Together they form a unique fingerprint.

Cite this