Virtual fundamental class

In mathematics, specifically enumerative geometry and symplectic geometry, the virtual fundamental class ${\displaystyle [X]^{\text{vir}}\in H_{*}(X)}$ of a (typically very singular) space ${\displaystyle X}$ (or a stack) is a generalization of the classical fundamental class of a smooth manifold which has better behavior with respect to the enumerative problems being considered. In this way, there exists a cycle with can be used for answering specific enumerative problems, such as the number of degree ${\displaystyle d}$ rational curves on a quintic threefold. For example, in Gromov–Witten theory, the Kontsevich moduli spaces

${\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,\beta )}$

for ${\displaystyle X}$ a smooth complex projective variety (or a symplectic manifold) ${\displaystyle \beta \in H_{2}(X)}$ a curve class, could have wild singularities such as^{pg 503} having higher-dimensional components at the boundary than on the main space. One such example is in the moduli space

${\displaystyle {\overline {\mathcal {M}}}_{1,n}(\mathbb {P} ^{2},1[H])}$

for ${\displaystyle H}$ the class of a line in ${\displaystyle \mathbb {P} ^{2}}$. The non-compact "smooth" component is empty, but the boundary contains maps of curves

${\displaystyle f:C\to \mathbb {P} ^{2}}$

whose components consist of one degree 3 curve which contracts to a point. There is a virtual fundamental class which can then be used to count the number of curves in this family.