The focus-focus addition graph is immersed

TL;DR

This paper studies the geometric structure of the addition graph in symplectic 4-manifolds with focus-focus singularities, showing it forms a Lagrangian immersion, which aids in constructing a symmetric monoidal structure on the Fukaya category.

Contribution

It proves that the closure of the focus-focus addition graph is a Lagrangian immersion, advancing understanding of symplectic fibrations with singularities.

Findings

The addition graph's closure is a Lagrangian immersion.

The geometry of the addition graph is characterized in the focus-focus case.

Results facilitate the construction of monoidal structures in Fukaya categories.

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.