(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
rational curves on a quintic threefold.
For example, in Gromov–Witten theory, the Kontsevich moduli spaces[3]
a smooth complex projective variety (or a symplectic manifold)
a curve class, could have wild singularities such as[4]pg 503 having higher-dimensional components at the boundary than on the main space.
The non-compact "smooth" component is empty, but the boundary contains maps of curves
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.
We can understand the motivation for the definition of the virtual fundamental class[5]pg 10 by considering what situation should be emulated for a simple case (such as a smooth complete intersection).
(representing the coarse space of some moduli problem
) which is cut out from an ambient smooth space
where it is transverse, then we can get a homology cycle by looking at the Euler class of the cokernel bundle
Now, this situation dealt with in Fulton-MacPherson intersection theory by looking at the induced cone
on the induced cone and the zero section, giving a cycle on
for which there is an embedding, we must rely upon deformation theory techniques to construct this cycle on the moduli space representing the fundamental class.
For the general case there is an exact sequence
Note the construction of Behrend-Fantechi is a dualization of the exact sequence given from the concrete example above[6]pg 44.
There are multiple definitions of virtual fundamental classes,[2][7][8][9] all of which are subsumed by the definition for morphisms of Deligne-Mumford stacks using the intrinsic normal cone and a perfect obstruction theory, but the first definitions are more amenable for constructing lower-brow examples for certain kinds of schemes, such as ones with components of varying dimension.
One of the first definitions of a virtual fundamental class[2]pg 10 is for the following case: suppose we have an embedding of a scheme
One natural candidate for such an obstruction bundle if given by
for the divisors associated to a non-zero set of generators
Then, we can construct the virtual fundamental class of
using the generalized Gysin morphism given by the composition
is the inverse of the flat pullback isomorphism
in the map since it corresponds to the zero section of vector bundle.
Then, the virtual fundamental class of the previous setup is defined as
which is just the generalized Gysin morphism of the fundamental class of
The first map in the definition of the Gysin morphism corresponds to specializing to the normal cone[10]pg 89, which is essentially the first part of the standard Gysin morphism, as defined in Fulton[10]pg 90.
In this way, the intermediate step of using the specialization of the normal cone only keeps the intersection-theoretic data of