Let $M$ be a simply-connected compact complex manifold of dimension three and $C$ is a smooth complex curve in $M$. Let $M'$ be the blow-up of $M$ along $C$.
My question is:
Is $M'$ also simply-connected?
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$\begingroup$ The question of birational invariance of the fundamental group has come up a few times on this website, but usually for algebraic varieties (where the answer is yes as long as everything is smooth and projective). I am fairly certain the answer should be positive in this case as well, but I don't actually know a reference (and I would be interested in finding a source that treats some version of this). $\endgroup$– R. van Dobben de BruynCommented Feb 5, 2023 at 17:10
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10$\begingroup$ That follows from the Seifert-van Kampen Theorem. Write $M$ as the union of $U=M\setminus C$ and $V$, a tubular neighborhood of $C$. Then $U\cap V$ is a punctured tubular neighborhood, i.e., a fiber bundle over $C$ whose fiber retracts onto a sphere $\mathbb{S}^{2c-1}$, where $c$ is the codimension of the smooth subvariety $C$ of $M$. There is analogous covering of the blowing up. Since $\mathbb{S}^{2c-1}$ is simply connected for $c>1$, the pushforward map from the fundamental group of $U\cap V$ to the fundamental group of $V$ is an isomorphism. $\endgroup$– Jason StarrCommented Feb 5, 2023 at 17:34
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