The focus-focus addition graph is immersed

Abstract

For a symplectic 4-manifold $M$ equipped with a singular Lagrangian fibration with a section, the natural fiberwise addition given by the local Hamiltonian flow is well-defined on the regular points. We prove, in the case that the singularities are of focus-focus type, that the closure of the corresponding addition graph is the image of a Lagrangian immersion in $(M \times M)^- \times M$, and we study its geometry. Our main motivation for this result is the construction of a symmetric monoidal structure on the Fukaya category of such a manifold.