There is a well-known fact (see for example HKLR) that if $p\colon Z\to \mathbb CP^1$ is a holomorphic fiber bundle admitting a holomorphic section $s\colon \mathbb CP^1\to Z$ such that $s^*N\cong \mathcal O(1)^{\oplus 2n}$ where $N$ is the normal bundle then the space $M$ of sections with such a normal bundle is of complex dimension $4n$ and admits a biquaternionic structure i.e. the action of $Mat_2(\mathbb C)$ on $TM$. If $Z$ is eqipped with an antiholomorphic involution $\tau$ restricting to the standard antiholomorphic involution on $\mathbb CP^1$ when the space $M^\tau$ of $\tau$-invartiant sections admits a natural hypercomplex structure. In this case one can reconstruct $Z$ from $M^\tau$ as its twistor space.

I'll remind how one gets the biquaternionic structure. Kodaira-Spencer tells us that $T_sM\cong H^0(s,N)$. There is also a vector bundle $E$ on $M$ such that $E_s\cong H^0(s,N(-1))$. We have a natural morphism $\alpha\colon E\otimes \mathbb C^2\to TM$ which will be an isomorphism if and only if $N$ is a direct sum of $\mathcal O(1)$. The tautological action of $Mat_2(\mathbb C)$ on $\mathbb C^2$ gives us an action on $TM$.

I've constructed a holomorphic fiber bundle $p\colon Z\to \mathbb CP^1$ with a $4n$-dimensional family of holomorphic sections with normal bundle $\mathcal O^{\oplus n}\oplus \mathcal O(2)^{\oplus n}$. Do spaces of sections of such fiber bundles admit any interesting structures? I understand that the subbundle of $TM$ corresponding to $H^0(\mathcal O(2)^{\oplus n})$ admits a natural action of $Mat_3(\mathbb C)$. And even more interesting question

Is there some way to reconstruct the given fiber bundle from the space of its holomorphic sections if the sections with normal bundle $\mathcal O(1)^{\oplus 2n}$ do not exist?

By the way, I don't think that there exist an antiholomorphic involution on my fiber bundle (only with essential singularities on $p^{-1}(\{0,\infty\})$) but there exists a holomorphic one.


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