8 added 1 characters in body

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 $\mathbb{F}_n = \mathbb{P}(\mathcal{O} \oplus \mathcal{O}(n)$.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}$. The uniqueness of the symplectic structrure in each case is a more difficult story, and was proven by Lalonde and McDuff [J-curves and the classification of rational and ruled symplectic 4-manifolds, Contact and Symplectic Geometry (Cambridge, 1994)]. 7 added 47 characters in body In order to obtain an explicit description of the diffeomorphism, one can use the following argument. Take$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{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)].

6 deleted 4 characters in body

In order to obtain an explicit description of the diffeomorphism, one can work as followsuse 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

$\mathbb{F}n$ and

$F{n-2k}$ are 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$\mathbb{F}_n$. 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 much 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)].

5 added 5 characters in body; added 12 characters in body; added 1 characters in body
Post Undeleted by Francesco Polizzi
4 added 878 characters in body; deleted 9 characters in body
3 added 23 characters in body
2 deleted 30 characters in body; added 5 characters in body; added 34 characters in body
Post Deleted by Francesco Polizzi
1