# Proper modifications of $\mathbb{C}^{n}$

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?

• 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 $\mathbb{C}^n$? – abx Sep 2 '18 at 3:26
• 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... – Praphulla Koushik Sep 2 '18 at 6:20
• 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. – Sasha Sep 2 '18 at 13:33
• What about blowing up $\Bbb{C}^n$ along a closed submanifold of positive dimension? – abx Sep 3 '18 at 4:15
• 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? – Lucas Kaufmann Sep 3 '18 at 8:30

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.