---
created: 2022-04-05T23:42
updated: 2023-12-16T00:00
aliases:
  - virtual fundamental cycle
  - virtual class
  - virtual cycle
---
See also:

- [Gromov-Witten](Gromov-Witten%20invariants.md)

# virtual fundamental class


$[X/Y] = f\shriek \in H_{2d}^{\BM}(X/Y; \QQ(d))$ where $1\in H_0^{\BM}(Y/Y; \QQ)$. See [Borel-Moore cohomology](Borel-Moore%20cohomology).


![](attachments/Pasted%20image%2020220319213951.png)

![](attachments/Pasted%20image%2020220325225635.png)
![](attachments/Pasted%20image%2020220325225849.png)

See [Alexander duality](Alexander%20duality), [[Unsorted/Kuranishi charts]].

![](attachments/Pasted%20image%2020220806144719.png)
![](attachments/Pasted%20image%2020220806144730.png)

# For moduli spaces

![](attachments/Pasted%20image%2020220325230405.png)
![](attachments/Pasted%20image%2020220325230416.png)

See construction of global [[Kuranishi charts]].