---
tag: 56V4WT0BT2
kind: definition
name: Modification of a complex space
msc: [32S45, 32H35]
defines: SB3MY7XQM9
uses: [831N2Q4GCB, 15XEBGNGHC, GB7M582M5Y, 0VQYCCQVJ5, C4KGYAFPN7]
sources:
  - url: https://en.wikipedia.org/wiki/Proper_morphism
---

## Definition

Let $X$ and $Y$ be [[reduced complex spaces|831N2Q4GCB]].
A holomorphic map $f \colon X \to Y$ is a *modification* if:

1. $f$ is [[proper|15XEBGNGHC]] and surjective;
2. there is a [[nowhere dense|0VQYCCQVJ5]] [[analytic subset|GB7M582M5Y]] $B \subseteq Y$ such that $f^{-1}(B)$ is nowhere dense in $X$;
3. $f$ restricts to a biholomorphic map $X \setminus f^{-1}(B) \to Y \setminus B$.

## Remarks

Example: the [[blow-up|C4KGYAFPN7]] $f \colon X \to \C^2$ of $\C^2$ at the origin is a modification with $B = \set{0}$.
It is proper and surjective, $f^{-1}(0)$ is a projective line, which is nowhere dense in the $2$-dimensional manifold $X$, and $f$ is biholomorphic off it.

Non-example: let $X$ be the disjoint union of $\C$ and a point $p$, and let $f \colon X \to \C$ be the identity on $\C$ and send $p$ to $0$.
Then $f$ is proper and surjective, and with $B = \set{0}$ clause 3 holds.
But $f^{-1}(B)$ contains the isolated point $p$, so it is not nowhere dense and clause 2 fails.
No other choice of $B$ helps, since $f$ is not injective over $0$, so $p$ must lie over $B$.

The set $f^{-1}(B)$ is the exceptional set.
Some authors call these maps proper modifications.