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.



Why are the following varieties symplectomorphic?
In order to obtain an explicit description of the diffeomorphism, one can use the following argument.


Take $B=\mathbb{C}^{n-1}$, with coordinates $t\_1, \ldots, t\_{n-1}$, and consider the complex space
$\mathcal{X}$ obtained glueing $\mathbb{P}^1 \times \mathbb{C} \times B$ with $\mathbb{P}^1 \times \mathbb{C} \times B$ by the identification of 


$(y\_0, y\_1, z, t\_1, \ldots, t\_{n-1})$ with $(y\_0', y\_1', z',t\_1, \ldots, t\_{n-1})$


if 


$z'=z^{-1}, \quad y\_1'=y\_1z^{-n}, \quad y\_0'=y\_0+y\_1 \sum\_{i=1}^{n-1}t\_iz^{-i}$.


Let us denote by $\pi \colon \mathcal{X} \to B$ the family obtained in this way.


Let now $T\_k \subset B$ be the determinantal locus given by rank $M \leq k$, where $M$ is the matrix


\begin{bmatrix}
t\_1 & \ldots & t\_{k+1} \cr
t\_2 & \ldots & t\_{k+2} \cr
\cdot & \cdot & \cdot \cr
\cdot & \cdot & \cdot \cr
\cdot & \cdot & \cdot \cr
t\_{n-k-1} & \ldots & t\_{n-1}
\end{bmatrix} 


Then, if $t \in T\_k - T\_{k-1}$ we have $X\_t :=\pi^{-1}(t) \cong \mathbb{F}\_{n-2k}$.


By using Ehresmann theorem, one concludes that 


$F\_n$ is diffeomorphic to $F\_{n-2k}$.


Geometrically speaking, we are considering all the rank $2$ vector bundles $V$ which fit into the short exact sequence


$0 \to \mathcal{O}\_{\mathbb{P}^1} \to V\_n \to \mathcal{O}\_{\mathbb{P}^1}(n) \to 0$.


They are classified by $H^1(\mathbb{P}^1, \mathcal{O}(-n)) \cong \mathbb{C}^{n-1}$, and we consider the family of ruled surfaces $\mathbb{P}(V\_n)$, thus obtained, as a deformation of $\mathbb{F}\_n = \mathbb{P}(\mathcal{O} \oplus \mathcal{O}(n))$.


This argument shows that even Hirzebruch surfaces are diffeomorphic to $S^2 \times S^2$, whereas odd Hirzebruch surfaces are diffeomorphic to $\mathbb{CP}^2 \sharp \overline{\mathbb{CP}^2}$.


The uniqueness of the symplectic structrure in each case is a more difficult story, and was proven by Lalonde and McDuff [J-curves and the classification of rational and
ruled symplectic 4-manifolds, Contact and Symplectic Geometry (Cambridge,
1994)].


