We give a generalization of toric symplectic geometry to Poisson manifolds which are
symplectic away from a collection of hypersurfaces forming a normal crossing configuration.
We introduce the tropical momentum map, which takes values in a generalization of affine
space called a log affine manifold. Using this momentum map, we obtain a complete
classification of such manifolds in terms of decorated log affine polytopes, hence extending
the classification of symplectic toric manifolds achieved by Atiyah, Guillemin-Sternberg …

Toric symplectic geometry was revolutionized in the 1970s and 1980s by Atiyah, Guillemin-
Sternberg, Kostant, and Delzant [1, 9, 17, 5], who essentially proved that toric symplectic
manifolds are encoded combinatorially by a rational polytope, which is the image of the
classical momentum map associated to the toric action (this is usually referred to as the
Delzant correspondence). In the present paper, we present a generalization of toric
symplectic geometry to a class of Poisson manifolds, called log symplectic manifolds, which …