---
tag: M5EK9AKXNR
kind: concept
name: Vanishing cycle
msc: [32S30, 14D05, 32S50]
sources:
  - url: https://en.wikipedia.org/wiki/Vanishing_cycle
same_as:
  wikidata: Q4205180
  wikipedia: Vanishing cycle
---

## Summary

A vanishing cycle is a homology class in a smooth fibre of a holomorphic function, see the [[definition|QEZFT0EM0G]].
It shrinks to a point as the fibre approaches a critical fibre.
Near a nondegenerate critical point it is represented by an embedded sphere of the middle dimension.
Vanishing cycles generate the homology of the [[Milnor fibre|X5HMTTGB62]] of an isolated singularity, and [[Picard–Lefschetz theory|FFYQAVT7MB]] computes the monodromy in terms of them.
The idea was later turned into the vanishing cycle functor on sheaves, used in étale cohomology and in the theory of [[perverse sheaves|5WHXRZDZFK]].

## Explanations

### L1 human

Consider the curves $x^2 + y^2 = t$ in $\C^2$ for small $t$.
For $t \ne 0$ each curve is a smooth tube, and it contains the circle of real points of radius $\sqrt{t}$ when $t > 0$.
As $t$ goes to $0$, this circle shrinks to the origin, where the curve becomes two crossing lines.
The circle is the vanishing cycle: a loop that disappears when the fibre becomes singular.