# Extend a vector bundle on a flat family

Let $f: X\to T$ be a flat family, and $\mathcal{F}_t$ is a vector bundle on $X_t$ for some $t\in T$. Can this $\mathcal{F}_t$ be extended to a vector bundle $\mathcal{F}$ on $f^{-1}(U)$ for some open neighborhood of $t$?

If moreover $\mathcal{E}$ is a vector bundle on $X$, and $\mathcal{F}_t$ is a subbundle of $\mathcal{E}_t$ on $X_t$, then can this $\mathcal{F}_t$ be extended to a subbundle $\mathcal{F}$ of $\mathcal{E}$ on $f^{-1}(U)$ for some open neighborhood of $t$?

• No, typically you cannot extend such a vector bundle. For instance, for $X\to T$ the family of all smooth degree $4$ surfaces in $\mathbb{P}^3$, for $t\in T$ such that $X_t$ contains a line $L$, the invertible sheaf $\mathcal{F}_t = \mathcal{O}_{X_t}(L)$ does not extend, even if you work with an analytic open neighborhood of $t$ in $T$. Nov 16 '15 at 19:47
• Excellent! But I just find an easy example, say there exists a family of vector bundles $F$ (say with parameter $t$) on $\mathbb{P}^1$ with such that $F_0=\mathcal{O}(1)\oplus \mathcal{O}(-1)$, but trivial for other $t$. What exactly is this? Nov 17 '15 at 1:35
• Yes, there is such an example. I believe I wrote about this before here. If you projectivize this, you obtain a family of Hirzebruch surfaces $\mathbb{P}^1 \times \mathbb{P}^1$ specializing to the Hirzebruch surface $\Sigma_2$. Nov 17 '15 at 9:57