Virtual fundamental class

Summary

A virtual fundamental class is a homology class that plays the role of the fundamental class. 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 and Donaldson–Thomas invariants.