---
tag: 5VXYQ4BXRB
kind: concept
name: Virtual fundamental class
msc: [53D45, 14N35, 57R95]
sources:
  - url: https://en.wikipedia.org/wiki/Virtual_fundamental_class
same_as:
  wikidata: Q108101855
  wikipedia: Virtual fundamental class
---

## Summary

A virtual fundamental class is a homology class that plays the role of the [[fundamental class|ZEA0PEM3B2]].
It is attached to a moduli space that need not be a manifold of the expected dimension.
It lives in the expected, or virtual, dimension: the number of unknowns minus the number of equations that cut out the space.
It is built by perturbing the equations to transverse ones.
In algebraic geometry it is built from a perfect obstruction theory, as done by Li and Tian and by Behrend and Fantechi.
Integrating cohomology classes against it defines invariants such as [[Gromov–Witten invariants|XVJA0JD79P]] and Donaldson–Thomas invariants.

## Explanations

### L1 human

When a space is cut out by equations, one expects its dimension to be the number of unknowns minus the number of equations.
Sometimes the equations are degenerate and the actual solution space is bigger, or singular.
The virtual fundamental class is what the solution space would contribute if the equations were in general position.
For example, the zero vector field on the sphere vanishes everywhere, so its zero set is the whole sphere.

A generic vector field on the sphere vanishes at points whose signed count is $2$.
The virtual class is these points counted with signs, which is $2$ times the class of a point.