---
date: 2022-04-22 23:34
modification date: Friday 22nd April 2022 23:34:06
title: "intersection cohomology"
aliases: [intersection cohomology, intersection homology]
---

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

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Hilbert scheme](Hilbert%20scheme.md)

---

# intersection cohomology

![](attachments/Pasted%20image%2020220422233408.png)


![](attachments/Pasted%20image%2020220422233451.png)
![](attachments/Pasted%20image%2020220422233503.png)

# Intersection complex

![](attachments/Pasted%20image%2020220422234017.png)
![](attachments/Pasted%20image%2020220422234111.png)

The $\IC^{-i}$ are [soft sheaves](soft%20sheaves).

Taking [hypercohomology](Unsorted/hypercohomology.md) computes the intersection cohomology:
![](attachments/Pasted%20image%2020220422234138.png)

These yield [perverse sheaves](Unsorted/perverse%20sheaf.md):
![](attachments/Pasted%20image%2020220422234157.png)

In terms of [local systems](local%20systems)
![](attachments/Pasted%20image%2020220422234221.png)

![](attachments/Pasted%20image%2020220422234310.png)

# Misc properties and results

![](attachments/Pasted%20image%2020220422234320.png)
![](attachments/Pasted%20image%2020220422234350.png)

A push-pull LES:
![](attachments/Pasted%20image%2020220422234509.png)

![](attachments/Pasted%20image%2020220422234547.png)

![](attachments/Pasted%20image%2020220422234650.png)



# Examples

![](attachments/Pasted%20image%2020220422233523.png)
![](attachments/Pasted%20image%2020220422233537.png)

![](attachments/Pasted%20image%2020220422233511.png)
![](attachments/Pasted%20image%2020220422233551.png)
![](attachments/Pasted%20image%2020220422233819.png)

![](attachments/Pasted%20image%2020220422234242.png)