Duistermaat-Heckman公式与多边形模空间的上同调
The Duistermaat-Heckman formula and the cohomology of moduli spaces of polygons
摘要：
本文给出了具有固定边长的空间多边形空间$M(r)$的上同调环$r \in \mathbb R^n_+$.这些空间可以被描述为$\mathbb C^n$中2-平面的Grassmanann通过乘法的$U(1)^n$-作用的辛约化，其中$U(1)^n$是酉群U（n）中对角矩阵的环面。证明了与纤维化$r$-水平集$\rightarrow M(r)$相关联的$n$线丛的第一类Chern类生成上同调环$H^* (M(r), \mathbb C).$。然后我们从这篇文章中推导出这些生成元的关系给出了描述$M(r).$体积的多项式函数，并给出了$M(r) $和$M(r')$之间的双有理映射的显式描述，$r$和$r'$位于矩多面体的不同腔室中。这种跨壁分析是证明上述陈类是$H^*(M(r))$的生成元的关键步骤（当$M(r)$是环面时，这是众所周知的，通过跨壁分析，我们证明当$M(r)$不是环面时它也成立）。
英文摘要：
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).
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