The Affine Invariant of Generalized Semitoric Systems

A generalized semitoric system F := (J,H) : M → R on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S-action and is not necessarily proper. These systems can exhibit focusfocus singularities, which correspond to fibers of F which are topologically multi-pinched tori. The image F (M) is a singular affine manifold which contains a distinguished set of isolated points in its interior: the focus-focus values {(xi, yi)} of F . By performing a vertical cutting procedure along the lines {x := xi}, we construct a homeomorphism f : F (M)→ f(F (M)), which restricts to an affine diffeomorphism away from these vertical lines, and generalizes a construction of Vũ Ngo.c. The set ∆ := f(F (M)) ⊂ R is a symplectic invariant of (M,ω, F ), which encodes the affine structure of F . Moreover, ∆ may be described as a countable union of planar regions of four distinct types, where each type is defined as the region bounded between the graphs of two functions with various properties (piecewise linear, continuous, convex, etc). If F is a toric system, ∆ is a convex polygon (as proven by Atiyah and Guillemin-Sternberg) and f is the identity.