## Journal of Symplectic Geometry

- J. Symplectic Geom.
- Volume 12, Number 1 (2014), 171-213.

### The Duistermaat-Heckman formula and the cohomology of moduli spaces of polygons

#### Abstract

We give a presentation of the cohomology ring of spatial polygon spaces $M(r)$ with fixed side lengths $r \in \mathbb{R}^n_+$. These spaces can be described as the symplectic reduction of the Grassmaniann of 2-planes in $\mathbb{C}^n$ by the $U(1)^n$-action by multiplication, where $U(1)^n$ is the torus of diagonal matrices in the unitary group $U(n)$. We prove that the first Chern classes of the $n$ line bundles associated with the fibration ($r$-level set) $\to M(r)$ generate the cohomology ring $H*(M(r),\mathbb{C})$. By applying the Duistermaat-Heckman Theorem, we then deduce the relations on these generators from the piece-wise polynomial function that describes the volume of $M(r)$. We also give an explicit description of the birational map between $M(r)$ and $M(r ')$ when the lengths vectors $r$ and $r '$ are in different chambers of the moment polytope. This wallcrossing analysis is the key step to prove that the Chern classes above are generators of $H*(M(r))$, (This is well-known when $M(r)$ is toric, and by wall-crossing we prove that it holds also when $M(r)$ is not toric).

#### Article information

**Source**

J. Symplectic Geom., Volume 12, Number 1 (2014), 171-213.

**Dates**

First available in Project Euclid: 29 August 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.jsg/1409317367

**Mathematical Reviews number (MathSciNet)**

MR3194079

**Zentralblatt MATH identifier**

1307.53072

#### Citation

Mandini, Alessia. The Duistermaat-Heckman formula and the cohomology of moduli spaces of polygons. J. Symplectic Geom. 12 (2014), no. 1, 171--213. https://projecteuclid.org/euclid.jsg/1409317367