A folded symplectic form on an even-dimensional manifold is a closed two-form that degenerates in a suitably controlled way along a smooth hypersurface. When a torus having half the dimension of the manifold acts in a way preserving the folded symplectic form and admitting a moment map, the manifold is called a folded symplectic toric manifold. Motivated by results in symplectic geometry, our goal is to prove a classification theorem of folded symplectic toric manifolds. This work is a step in that direction: the main result is a necessary and sufficient condition for two orientable, folded symplectic toric four-manifolds to be isomorphic.
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