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What is the symplectic form on the manifold whose associated Delzant polytope is a trapezoid? I am trying to find it by using the Marsden–Weinstein theorem, but I have been unable to do so. If someone knows the symplectic form exactly, please describe it or provide a reference.

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  • $\begingroup$ Maybe it would be helpful if you told us how you think about the above toric symplectic manifold associated to a trapezoid (b.t.w. those are called Hirzebruch surfaces). This will ensure that the answer you get is most useful to you. $\endgroup$ Jun 2, 2015 at 14:51
  • $\begingroup$ The torus action is (s,t)(z_0,z_1,z_2,z_3)=(st*z_0,s*z_1,s*z_2,t*z_3).I made a reduction space by using it and some text book said it can be idetifed to Hirzeburch surface by [z_0,z_1,z_2,z_3]→([z_1,z_2],[z_1*z_3,z_2*z_3,z_0]). $\endgroup$ Jun 2, 2015 at 15:57
  • $\begingroup$ and Hirzebruch surface has Kahler form induced by CP^1×CP^2,the book said.but I can't find the symplectic form by made from Delzant construction.Delzant construction is using morment map pull back a reguler and reduct it by torus action. I want to know my method is right or not. $\endgroup$ Jun 2, 2015 at 16:10
  • $\begingroup$ sorry for my bad groummer and mistaking spells $\endgroup$ Jun 2, 2015 at 16:11
  • $\begingroup$ I'd recommend that you have a look at Ana Cannas da Silva's Lectures on Symplectic Geometry. It contains a thorough exposition of Delzant's theorem and the reconstruction process. In fact, if I remember correctly, she explicitly computes the very examples that you are working on (trapezoids). $\endgroup$ Jun 4, 2015 at 13:39

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