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Let $f\colon X\to \mathbb P^1$ be a proper morphism of smooth complex algebraic varieties and let $p\in\mathbb P^1$. Are there a complex disk $\Delta\subseteq\mathbb P^1$ and a Zariski open subset $U\subseteq \mathbb P^1$, with $p\in\Delta\subseteq U$, such that $H^1(f^{-1}(U),{{\mathcal O}^{\rm an}}^*)\to H^1(f^{-1}(\Delta),{{\mathcal O}^{\rm an}}^*)$ is an isomorphisminjection?

Let $f\colon X\to \mathbb P^1$ be a proper morphism of smooth complex algebraic varieties and let $p\in\mathbb P^1$. Are there a complex disk $\Delta\subseteq\mathbb P^1$ and a Zariski open subset $U\subseteq \mathbb P^1$, with $p\in\Delta\subseteq U$, such that $H^1(f^{-1}(U),{{\mathcal O}^{\rm an}}^*)\to H^1(f^{-1}(\Delta),{{\mathcal O}^{\rm an}}^*)$ is an isomorphism?

Let $f\colon X\to \mathbb P^1$ be a proper morphism of smooth complex algebraic varieties and let $p\in\mathbb P^1$. Are there a complex disk $\Delta\subseteq\mathbb P^1$ and a Zariski open subset $U\subseteq \mathbb P^1$, with $p\in\Delta\subseteq U$, such that $H^1(f^{-1}(U),{{\mathcal O}^{\rm an}}^*)\to H^1(f^{-1}(\Delta),{{\mathcal O}^{\rm an}}^*)$ is an injection?

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bog
bog
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  • 1
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Analytic vs Zariski neighbourhood of a fibre

Let $f\colon X\to \mathbb P^1$ be a proper morphism of smooth complex algebraic varieties and let $p\in\mathbb P^1$. Are there a complex disk $\Delta\subseteq\mathbb P^1$ and a Zariski open subset $U\subseteq \mathbb P^1$, with $p\in\Delta\subseteq U$, such that $H^1(f^{-1}(U),{{\mathcal O}^{\rm an}}^*)\to H^1(f^{-1}(\Delta),{{\mathcal O}^{\rm an}}^*)$ is an isomorphism?