Abstract
We develop a localization theorem in equivariant cohomology that generalizes both the localization with respect to a component of the momentum map and the localization with respect to the norm square of the momentum map, in that it allows a more flexible polarization. Our proof uses noncompact cobordisms. Applying a case of our formula to symplectic toric manifolds gives the measure-theoretic version of the classical Brianchon-Gram polytope decomposition, thus answering a question posed by Shlomo Sternberg.
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