Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
The notion of virtual fundamental class is a generalization of that of fundamental class from manifolds to more general spaces in higher geometry, notably to orbifolds and their equivalent incarnations as stacks, as well as to derived manifolds.
This plays a central role when these stacks serve as moduli stacks for certain structures on some space (certain maps into that space regarded as a target space, notably) and one is interested in the relevant “path integral” over all these structures (to produce invariants of target space). This is manifestly so for instance in the application to Gromov-Witten invariants. In these cases the pairing of cocycles against the virtual fundamental class plays the role of integration over the given moduli stack.
Given a locally Noetherian derived scheme, $(X, \mathcal{O}_X)$ with underlying scheme $t_0X$, $\pi_i\mathcal{O}_X$ coherent, and $\pi_i\mathcal{O}_X = 0$ for $i\gg0$, the virtual fundamental class is defined by first constructing a class in $G_0(t_0X)$ (the K-theory of coherent sheaves) and then using this to produce an element of the Chow homology of $t_0X$. In our hearts we all know that the fundamental class should be something tautological, just like how the fundamental class of a triangulated manifold in simplicial homology is the manifold itself. We can use this idea once we have the following theorem:
(Devissage).
The natural map of derived schemes $j: t_0X \rightarrow X$ induces an isomorphism $j_*: G_0(t_0X) \rightarrow G_0(X)$.
A proof of this theorem and a definition of G-theory for derived schemes in (Barwick).
The $K$-theory virtual fundamental class of a derived scheme $(X, \mathcal{O}_X)$ is the unique element $[X]$ in $G_0(t_0X)$ such that $j_*[X] = [\mathcal{O}_X]$ where $j: t_0X \rightarrow X$ is the natural map.
If you trace through the definition of $G_0$ and the identification of the heart of $Coh(X)$ with $Coh(t_0X)$, you can show that $[X] = \sum (-1)^i [\pi_i\mathcal{O}_X]$.
To get from here to a virtual fundamental class in cohomology we need some more assumptions on $X$. For example, let’s assume $X$ is finitely-presented over a field $k$ and that $\mathbb{L}_{X/k}$ is concentrated in degrees $0$ and $-1$, locally up to quasi-isomorphism, and a perfect complex. Then $j^*\mathbb{L}_{X/k}$ is a perfect complex over $t_0X$ and the dual is called the virtual tangent sheaf $\mathbb{T}^{vir}$.
With the above assumptions and notation, the virtual fundamental class of $X$ is the evaluation of the inverse of the Todd class $Td(\mathbb{T}^{vir})$ on $\tau([X])$, where $[X]$ is as in def. 1 and $\tau$ is the Grothendieck-Riemann-Roch transformation.
Although the moduli space of stable maps is sometimes referred to as a compactifiaction of the space of maps, in analogy with the Deligne-Mumford compactification of the moduli space of curves, in fact it typically has boundary components of higher dimension than the space it was supposed to compactify!
Take for example $\bar M_{1,0}(P^2,3)$. It ought to be a compactification of the space of degree-3 maps from genus-1 curves to $P^2$, and indeed one of its components has a Zariski open subset birational to the $P^9$ of all plane cubics. But there is also a ‘boundary component’ of higher dimension, namely the boundary component consisting of maps whose domain is a genus-1 curve glued to a nodal rational curve: the nodal curve maps to a rational cubic in $P^2$, while the $g=1$ component contracts to a point on that nodal cubic.
This boundary component has dimension 10: namely, there are 8 parameters to specify the image nodal cubic, 1 paramenter to determine the point to which the $g=1$ component contracts, and finally there is 1 paramenter for the j-invariant for the $g=1$ component. The topological fundamental class lives in dimension 10 so it is rather useless to integrate against if all your cohomology classes are codimension 9 — which is the expected dimension.
The virtual fundamental class always lives in the expected dimension.
(The expected dimension is often the one you would expect(!) from naive counts like the above. More formally it can be computed as dim $H^0(C,N_f)$, where $f:C\to P^2$ is a moduli point (with normal bundle $N_f$) such that $H^1(C,N_f)=0$ (this is to say that the first order infinitesimal deformations are unobstructed).)
The situation is analogous (possibly in fact a special case of) the standard situation in intersection theory when a section of a vector bundle is not regular: its zero locus is then of too high dimension and is of little use to intersect against. The correct class to work with is then the top Chern class of the vector bundle (cf. [Fulton] ch.14), which could be called the virtual class of the zero locus.
In the example above, I don’t know right now if the virtual class in fact appears as a top Chern class of a vector bundle — I think it should, because the excess is just a variation of the standard example $\bart M_{1,1}(X,0)$, and in that example it is true that the virtual class appears as a top Chern class: there is a so-called obstruction bundle? which in this case is the dual of the Hodge bundle from the factor $\bar M_{1,1}$ tensored with the tangent bundle from $X$.
(The Hodge bundle is the direct image bundle of the canonical bundle of the universal curve, hence of rank $g$, hence just a line bundle in this case.)
The virtual fundamental class is the top Chern class of the obstruction bundle (cap the topological fundamental class).
In this case, $dim \bar M_{1,1}(X,0) = 1 + dim X$, and the obstruction bundle has rank $dim X$, hence the virtual class has dimension 1.
Perhaps it should be mentioned also that the moduli space of maps can have components of too high dimension even before it is ‘compactified’, and even without involving contracting curves. A famous example is $M_{0,0}(Q,d)$ (no bar needed for this argument) where $Q$ is a quintic three-fold. Let’s say $d=2$, so we are talking about conics on the quintic three-fold. Since $Q$ has trivial canonical class it follows that the expected dimension is always 0 (i.e. in every degree there ought to be a finite number of rational curves on Q).
But now, $M_{0,0}(Q,2)$ is a space of maps, not a space of curves, and for every one of the famous 2875 lines on $Q$ there is a 2-dimensional family of double covers of the line, which clearly count as stable degree-2 maps, so $M_{0,0}(Q,2)$ contains 2875 components of dimension 2, in contrast to the virtual dimension 0.
Virtual fundamental classes play a central role in the theory of Gromov-Witten invariants.
The idea of virtual fundamental classes and corresponding picture of derived moduli spaces comes from
A quick overview of virtual fundamental classes for algebraic stacks is in section 4.4.3 of
A more detailed overview with many pointers to the literature is in section 8 “Introduction to the virtual fundamental class” of
A review of virtual fundamental classes of orbifolds in differential geometry is around page 7 of
Detailed disucssion for the branched manifold-variant of orbifolds is in section 3.4 of
Virtual fundamental classs of derived manifolds are discussed in section 13.2 of
with applications to moduli spaces of coherent sheaves on Calabi-Yau 4-folds in
Specifically for applications to Gromov-Witten theory see for instance
The material in Moduli space of stable maps above originates in a blog discussion here
The definition and description above for derived schemes derives from p.23 of
The devissage result needed is proven as Proposition 8.2 here: