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March 2014 The Duistermaat-Heckman formula and the cohomology of moduli spaces of polygons
Alessia Mandini
J. Symplectic Geom. 12(1): 171-213 (March 2014).

Abstract

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) $\to M(r)$ 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 wallcrossing 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).

Citation

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Alessia Mandini. "The Duistermaat-Heckman formula and the cohomology of moduli spaces of polygons." J. Symplectic Geom. 12 (1) 171 - 213, March 2014.

Information

Published: March 2014
First available in Project Euclid: 29 August 2014

zbMATH: 1307.53072
MathSciNet: MR3194079

Rights: Copyright © 2014 International Press of Boston

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Vol.12 • No. 1 • March 2014
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