2008/11/25 by Alessia Mandini, Mandini, Alessia
Mathematics · #14F40 #53D30 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.0811.4062
openalex publication_date 2008/11/25 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We give a presentation of the cohomology ring of spatial polygon spaces\nM(r) with fixed side lengths r \∈ mathbb Rn+. These spaces can be\ndescribed as the symplectic reduction of the Grassmaniann of 2-planes in\n mathbb Cn by the U(1)n-action by multiplication, where U(1)n is the\ntorus of diagonal matrices in the unitary group U(n). We prove that the first\nChern classes of the n line bundles associated with the fibration r-level\nset \→ M(r) generate the cohomology ring H^* (M(r), mathbb C). By\napplying the Duistermaat--Heckman Theorem, we then deduce the relations on\nthese generators from the piece-wise polynomial function that describes the\nvolume of M(r). We also give an explicit description of the birational map\nbetween M(r) and M(r') when the lengths vectors r and r' are in\ndifferent chambers of the moment polytope. This wall-crossing analysis is the\nkey step to prove that the Chern classes above are generators of H^*(M(r))\n(this is well-known when M(r) is toric, and by wall-crossing we prove that it\nholds also when M(r) is not toric).\n