Monodromy of a punctured disc

Question

I meet the following problem which I think related to the monodromy:

Let $D: = \{z \mid |z|<1 \}$ be a disc, and $U \to D$ be a variety fibred over $D$. For each point $t \in D \backslash \{0\}$, the fibre $U_t$ is isomorphic to a fixed variety $X$, then is $U$ isomorphic to the product space $D \times X$?

I think when $X$ is a projective space $\mathbb{P}^n$, the result is true.

Any suggestions on the problem/references are great welcome!!

Edit:

Since this is not true in general, is it true that $U$ and $D \times X$ are birational?

Answer 1

This is well-known to be false. A classical example is given by the Hirzebruch surfaces $\mathbb{F}_n$. For instance, consider on $\mathbb{P}^1$ all extensions $$0\rightarrow \mathscr{O}_{\mathbb{P}^1}(-1)\rightarrow E_e \rightarrow\mathscr{O}_{\mathbb{P}^1}(1)\rightarrow 0\ .$$These extensions are parametrized by a class $e\in \mathrm{Ext}^1(\mathscr{O}_{\mathbb{P}^1}(1),\mathscr{O}_{\mathbb{P}^1}(-1))\cong \mathbb{C}$; for $e\neq 0$ it is easy to see that $E_e\cong \mathscr{O}_{\mathbb{P}^1}^2$, while $E_0=\mathscr{O}_{\mathbb{P}^1}(-1)\oplus \mathscr{O}_{\mathbb{P}^1}(1)$. Now consider the corresponding family of projective bundles over $\mathbb{C}$; for $e\neq 0$ you get $\mathbb{P}^1\times \mathbb{P}^1$, while the fiber at $0$ is the surface $\mathbb{F}_2$.