I look forward very much to hear more details about the derived

viewpoint on virtual classes!

Meanwhile here are some down-to-earth geometric examples (plane cubics,

and conics on the quintic three-fold) to illustrate the need of the

virtual fundamental class.

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 Mbar_{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

Mbar_{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 Mbar_{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 Mbar_{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.

Cheers,

Joachim.

## Re: A Seminar on Gromov-Witten Theory

Hello Urs; I just went through and cleaned it up a bit and added a few things…

And actually the seminar is being run by me (also from Berkeley, but not a Teichner student) and two other MPI grad students.