Mathematics – Symplectic Geometry
Scientific paper
2008-11-25
Mathematics
Symplectic Geometry
31pages, 2 figures
Scientific paper
We give a presentation of the cohomology ring of polygon spaces $M(r)$ with fixed side length $r \in \R^n_+$. These spaces can be described as the symplectic reduction of the Grassmaniann of 2-planes in $\C^n$ 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 $ \textrm{($r$-level set)} \rightarrow M(r)$ generate the cohomology ring $H^* (M(r), \Q).$ 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 length 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).
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