Act globally, compute locally: group actions, fixed points, and localization

TL;DR

This paper explores the use of localization techniques in symplectic and algebraic geometry, demonstrating how global equivariant computations can be reduced to local fixed point data, with applications to toric varieties.

Contribution

It provides an expository overview of localization methods at the intersection of symplectic geometry, algebraic geometry, combinatorics, and representation theory, emphasizing toric techniques.

Findings

Localization simplifies global computations to fixed point data.

Examples from symplectic geometry and toric varieties illustrate the techniques.

The article highlights the interplay between topology, combinatorics, and geometry.

Abstract

Localization is a topological technique that allows us to make global equivariant computations in terms of local data at the fixed points. For example, we may compute a global integral by summing integrals at each of the fixed points. Or, if we know that the global integral is zero, we conclude that the sum of the local integrals is zero. This often turns topological questions into combinatorial ones and vice versa. This expository article features several instances of localization that occur at the crossroads of symplectic and algebraic geometry on the one hand, and combinatorics and representation theory on the other. The examples come largely from the symplectic category, with particular attention to toric varieties. In the spirit of the 2006 International Conference on Toric Topology at Osaka City University, the main goal of this exposition is to exhibit toric techniques that arise in symplectic geometry.