TITLE: Why are the following varieties symplectomorphic?
QUESTION [7 upvotes]: I saw a statement somewhere that for the Hirzebruch surfaces $F_n:=\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}\oplus\mathcal{O}(n))$, $F_n$ and $F_m$ are symplectormorphic when $m$ and $n$ have the same parity. 
My question is: Why is this true?
I can see that they are diffeomorphic by Freedman's Theorem: computing the intersection pairing on $\text{Pic}(F_n)=H^2(F_n, \mathbb{Z})$, which is an even form when n is even, and an odd form when n is odd.
But this result is deep and abstract, is there any easy way to construct a symplectomorphism? Certainly it is not given by polynomial maps, I'm wondering what it will be.

REPLY [2 votes]: I think that $F(n)$ with a Kahler form such that the base $\mathbb{P}^1$ has area 2 and 
the fibre has area 1, is symplectomorphic to $F(n+2)$ with the Kahler form such that base and fibre have area 1. 
There is a "visual proof" of this which uses an idea of Leung and Symington (in an article called "Almost toric manifolds", on the arxiv). The operation is called "nodal trade", and is pictured below. 
Figure A) is the image of the moment map of the toric variety $F(n)$. You can perturb the moment map to another Lagrangian fibration, so to replace a "corner" with a singular Lagrangian fibre (a pinched torus) over the red cross in figure B). Then you can slide this fibre along the dashed straight line until it hits the opposite edge and becomes another "corner". This gives figure C) which is the polytope of $F(n+2)$. 
Leung and Symington explain these operations carefully in the above paper.
alt text http://www.freeimagehosting.net/uploads/a5a7b4ad8e.jpg