Localization for equivariant cohomology with varying polarization

TL;DR

This paper generalizes and unifies various localization theorems in equivariant symplectic geometry, allowing for broader applications including degenerate forms and providing solutions to longstanding questions.

Contribution

It introduces a flexible polarization concept and a unified localization framework that encompasses previous theories and extends their applicability.

Findings

Unified localization formulas for equivariant cohomology.

Applicable to degenerate 2-forms and noncompact cobordisms.

Provided a solution to Sternberg's question on Brianchon-Gram decomposition.

Abstract

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 formulas are also valid for closed 2-forms 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.