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Let $f:X\to Y$ be a proper surjective holomorphic map between two $n$-dimensional connected complex manifolds $X$ and $Y$. $X$ is called a proper modification of $Y$ if there are nowhere dense compact analytic subsets $E\subset X$ and $S\subset Y$ such that the following hold:

(1) $f(E)\subset S$. (2) $f$ maps $X\setminus E$ biholomorphically onto $Y\setminus S$. (3) Every fibre $f^{-1}(y)$, $y\in S$, consists of more than one point.

Let $f:X\to\mathbb{C}^{n}$ be a proper modification of $\mathbb{C}^{n}$. Is $X$ necessarily an iterated blowup of $\mathbb{C}^{n}$ at finitely many points?

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  • $\begingroup$ I wonder what you call the Remmert reduction. If you take $M=X\times \mathbb{C}^n$, where $X$ is compact, isn't the Remmert reduction $\mathbb{C}^n$? $\endgroup$ – abx Sep 2 '18 at 3:26
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    $\begingroup$ do you want to say one/two lines about what is proper modification?? I hear the word blowup occasionally but did not hear proper modification.. Google search also does not say much... $\endgroup$ – Praphulla Koushik Sep 2 '18 at 6:20
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    $\begingroup$ Also it is better to make clear what do you mean by a blowup --- a blowup of a smooth subvariety, or a blowup of a sheaf of ideals. $\endgroup$ – Sasha Sep 2 '18 at 13:33
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    $\begingroup$ What about blowing up $\Bbb{C}^n$ along a closed submanifold of positive dimension? $\endgroup$ – abx Sep 3 '18 at 4:15
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    $\begingroup$ What if you blow up a point in $\mathbb C^n$ and then blow up a compact submanifold contained in the exceptional divisor of the first blow up? $\endgroup$ – Lucas Kaufmann Sep 3 '18 at 8:30
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No. A counterexample to your question is given by a blow-up at point in $\mathbb C^n$ followed by a blow-up along a compact submanifold contained in the exceptional divisor of the first blow-up.

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