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 ∈ Rn+. These spaces can be described as the symplectic reduction of the Grassmaniann of 2-planes in Cn 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).