The duistermaat-heckman formula and the cohomology of moduli spaces of polygons

We give a presentation of the cohomology ring of spatial polygon spaces M(r) with fixed side lengths r is an element of R-+(n). These spaces can be described as the symplectic reduction of the Grassmaniann of 2-planes in 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) -> M(r) generate the cohomology ring H*(M(r), 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 wall-crossing 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).