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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 relat"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0811.4062","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2008-11-25T11:47:10Z","cross_cats_sorted":[],"title_canon_sha256":"978ce045f66897a1f54176a062b9fa8668291b26ed4e3b35cd9fe0c48a3aa336","abstract_canon_sha256":"99a68d6a19038bf1de763ea2b412e95e74f4903d0b18cf650ee1008a030e2be3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:16:06.818192Z","signature_b64":"AaY9wZZSk0/5ohj84dK80UVdBv75UduGUohz9ZZCJm5Y2NLT2C6eEkS3igmVRxoX/KL0gYSnDC8xwe8Uxq6KAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1a97a3a7df8ed1baf8c7473d5fc89a0309c03d1c027d9a258a723d08627e774","last_reissued_at":"2026-05-18T03:16:06.817627Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:16:06.817627Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Duistermaat-Heckman formula and the cohomology of moduli spaces of polygons","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SG","authors_text":"Alessia Mandini","submitted_at":"2008-11-25T11:47:10Z","abstract_excerpt":"We give a presentation of the cohomology ring of spatial polygon spaces $M(r)$ with fixed side lengths $r \\in \\mathbb R^n_+$. 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