---
tag: SB3MY7XQM9
kind: concept
name: Modification of a complex space
msc: [32S45, 32H35]
sources:
  - url: https://en.wikipedia.org/wiki/Proper_morphism
---

## Summary

A modification of a [[reduced complex space|831N2Q4GCB]] $Y$ is a [[proper|15XEBGNGHC]] surjective holomorphic map $X \to Y$ that is biholomorphic outside thin analytic subsets.
It changes $Y$ only over a nowhere dense analytic set, where it may replace points by larger sets.
[[Blowing up|C4KGYAFPN7]], normalization and resolution of singularities give modifications.
Modifications are the analytic counterpart of proper birational morphisms in algebraic geometry.

## Explanations

### L1 human

A modification keeps a space as it is except along a small bad set, where it may replace points by curves or other larger pieces.
The basic example is the blow-up of the plane $\C^2$ at the origin.
It keeps every point except the origin, and it replaces the origin by a projective line that records the directions of lines through the origin.