---
date: 2022-04-09 21:56
modification date: Saturday 9th April 2022 21:56:11
title: "vanishing cycles"
aliases: [vanishing cycles, nearby cycles, vanishing cycle, nearby cycle, monodrom operator]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #todo/untagged
- Refs:
	- Motivic nearby cycles: <http://user.math.uzh.ch/ayoub/PDF-Files/Leiden.pdf#page=1>
- Links:
	- [Unsorted/lefschetz pencil](Unsorted/lefschetz%20pencil.md)
	- [nearby cycle functor](nearby%20cycle%20functor.md)
	- [Picard-Lefchetz](Unsorted/Picard%20Lefchetz%20theory.md)
	- [VHS](Unsorted/VHS.md)

---

# vanishing cycles

![](attachments/2023-03-12nearby.png)

![](attachments/Pasted%20image%2020220317210911.png)

![](attachments/Pasted%20image%2020220501005840.png)

# Nearby cycles

![](attachments/Pasted%20image%2020220413144917.png)

# Nearby/vanishing cycle complexes

![](attachments/2023-03-08nearby1.png)
![](attachments/2023-03-08saito.png)

# Misc

![](attachments/Pasted%20image%2020220422214353.png)


# Applications

![](attachments/Pasted%20image%2020220422214508.png)
![](attachments/Pasted%20image%2020220422214501.png)


## In Deligne's proof of Weil

![](attachments/Pasted%20image%2020220424140533.png)

# Definitions of complexes

![](attachments/Pasted%20image%2020220424141427.png)

![](attachments/Pasted%20image%2020220424141434.png)
![](attachments/Pasted%20image%2020220424141453.png)
![](attachments/Pasted%20image%2020220424141507.png)
![](attachments/Pasted%20image%2020220424141526.png)

Variation:
![](attachments/Pasted%20image%2020220424141609.png)

![](attachments/Pasted%20image%2020220424141623.png)
![](attachments/Pasted%20image%2020220424141641.png)

# Monodromy operator

![](attachments/2023-03-12mono.png)