In mathematics, specifically enumerative geometry and symplectic geometry, the virtual fundamental class [ X ] vir ∈ H ∗ ( X ) {\displaystyle [X]^{\text{vir}}\in H_{*}(X)} of a (typically very singular) space X {\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 d {\displaystyle d} rational curves on a quintic threefold. For example, in Gromov–Witten theory, the Kontsevich moduli spaces M ¯ g , n ( X , β ) {\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,\beta )} for X {\displaystyle X} a smooth complex projective variety (or a symplectic manifold) β ∈ H 2 ( X ) {\displaystyle \beta \in H_{2}(X)} a curve class, could have wild singularities such aspg 503 having higher-dimensional components at the boundary than on the main space. One such example is in the moduli space M ¯ 1 , n ( P 2 , 1 [ H ] ) {\displaystyle {\overline {\mathcal {M}}}_{1,n}(\mathbb {P} ^{2},1[H])} for H {\displaystyle H} the class of a line in P 2 {\displaystyle \mathbb {P} ^{2}} . The non-compact "smooth" component is empty, but the boundary contains maps of curves f : C → P 2 {\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.
Geometric motivation We can understand the motivation for the definition of the virtual fundamental classpg 10 by considering what situation should be emulated for a simple case (such as a smooth complete intersection). Suppose we have a variety X {\displaystyle X} (representing the coarse space of some moduli problem X {\displaystyle {\mathcal {X}}} ) which is cut out from an ambient smooth space Y {\displaystyle Y} by a section s {\displaystyle s} of a rank- r {\displaystyle r} vector bundle E → Y {\displaystyle E\to Y} . Then X {\displaystyle X} has "virtual dimension" ( n − r ) {\displaystyle (n-r)} (where n {\displaystyle n} is the dimension of Y {\displaystyle Y} ). This is the case if s {\displaystyle s} is a transverse section, but if s {\displaystyle s} is not, and it lies within a sub-bundle E ′ ⊂ E {\displaystyle E'\subset E} where it is transverse, then we can get a homology cycle by looking at the Euler class of the cokernel bundle E / E ′ {\displaystyle E/E'} over X {\displaystyle X} . This bundle acts as the normal bundle of X {\displaystyle X} in Y {\displaystyle Y} .
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