nLab
virtual fundamental class

Idea and examples

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 ${\overline{M}}_{1,0}(P^2,3)$. 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 $bart{M}_{1,1}(X,0)$ mentioned by A.J., 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 ${\overline{M}}_{1,1}$ 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, $\dim {\overline{M}}_{1,1}(X,0)=1+\dim X$, 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.

Related entries

• fundamental class

• Virtual fundamental classes play a central role in the theory of Gromov-Witten invariants.

References

The above material originates in a blog discussion here

• Ch. Okonek, A. Teleman, Computing virtual fundamental classes: gauge theoretical Gromov-Witten invariants for toric varieties (arXiv:math/0205137)