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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 bedescribed 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 thetorus of diagonal matrices in the unitary group U(n). We prove that the firstChern classes of the $n$ line bundles associated with the fibration $r$-levelset $\rightarrow M(r)$ generate the cohomology ring $H^* (M(r), \mathbb C).$ Byapplying the Duistermaat--Heckman Theorem, we then deduce the relations onthese generators from the piece-wise polynomial function that describes thevolume of $M(r).$ We also give an explicit description of the birational mapbetween $M(r) $ and $M(r')$ when the lengths vectors $r$ and $r'$ are indifferent chambers of the moment polytope. This wall-crossing analysis is thekey 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 itholds also when $M(r)$ is not toric).

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