This thesis begins with a generalization of two classic results about toric varieties to the context of varieties with codimension-one torus actions: the toric cone theorem and the intersection formula for invariant curves and divisors. Leaving algebraic toric geometry for symplectic toric geometry, we then prove a Delzant-type classification of torus actions on manifolds equipped with symplectic structures that have mild order-one singularities (called b-symplectic or log-symplectic manifolds). We end by studying symplectic structures with higher order singularities and updating theorems of Mazzeo, Melrose, and Radko from b-symplectic geometry to this more general context.
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Torus Actions and Singularities in Symplectic Geometry.