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

• Published in: arXiv.org 2013-08
• Main Author: Mandini, Alessia
• Format: Article
• Language: English
• Subjects: Generators; Geometry; Homology; Multiplication; Polynomials; Toruses
• ISSN: 2331-8422

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 \(\rightarrow M(r)\) 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 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).