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  "body": "\n## Summary\n\nA virtual fundamental class is a homology class that plays the role of the [[fundamental class|ZEA0PEM3B2]].\nIt is attached to a moduli space that need not be a manifold of the expected dimension.\nIt lives in the expected, or virtual, dimension: the number of unknowns minus the number of equations that cut out the space.\nIt is built by perturbing the equations to transverse ones.\nIn algebraic geometry it is built from a perfect obstruction theory, as done by Li and Tian and by Behrend and Fantechi.\nIntegrating cohomology classes against it defines invariants such as [[Gromov–Witten invariants|XVJA0JD79P]] and Donaldson–Thomas invariants.\n\n## Explanations\n\n### L1 human\n\nWhen a space is cut out by equations, one expects its dimension to be the number of unknowns minus the number of equations.\nSometimes the equations are degenerate and the actual solution space is bigger, or singular.\nThe virtual fundamental class is what the solution space would contribute if the equations were in general position.\nFor example, the zero vector field on the sphere vanishes everywhere, so its zero set is the whole sphere.\n\nA generic vector field on the sphere vanishes at points whose signed count is $2$.\nThe virtual class is these points counted with signs, which is $2$ times the class of a point.\n\n",
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