Localization for equivariant cohomology with varying polarization

The main contribution of this paper is a generalization of several previous localization theories in equivariant symplectic geometry, including the classical Atiyah-Bott/Berline-Vergne localization theorem, as well as many cases of the localization via the norm-square of the momentum map as initiated and developed by Witten, Paradan, and Woodward. Our version unifies and generalizes these theories by using noncompact cobordisms as in previous work of Guillemin, Ginzburg, and Karshon, and by introducing a more flexible notion of "polarization" than in previous theories. Our localization formulae are also valid for closed 2-forms ω that may be degenerate. As a corollary, we are able to answer a question posed some time ago by Shlomo Sternberg concerning the classical Brianchon-Gram polytope decomposition. We illustrate our theory using concrete examples motivated by our answer to Sternberg's question.