Ok, sorry this took so long, but I’m back. Hopefully will be more regular now. Anyway, last time we talked about Stable Maps, and in the meantime, there’s been a post at the Secret Blogging Seminar talking about Gromov-Witten invariants and TQFTs. We’re following a series of talks I gave here, so we’re not going in that direction at all. Instead, here I’m going to define G-W invariants for nice spaces, prove a couple of nice lemmas, and eventually get to using them to solve some actual enumerative problems (give me a few more posts before that happens, though.)
Anyway, for the rest of this series, we’ll use to denote a homogeneous variety. By this, I mean a variety which has transitive automorphism group. Equivalently, this is the variety of cosets of an algebraic group by a subgroup (to get between them, consider modulo the stabilizer of a point). These are generally nice spaces. For one thing, they’re always convex. Among the examples are algebraic groups, , flag varieties, and Grassmannians. We’ll also only be concerned with rational curves, because they behave MUCH better than higher genus curves.
For notational convenience, we set and , these being the standard topological homology and cohomology groups. We’ll also use the notation of integration for cap product with the fundamental class (thanks to the theorems from last time, there’s actually a fundamental class, so we don’t need to worry about virtual fundamental classes for this series…we’ll get into that some other time).
Now, to define the Gromov-Witten invariants, since we’ve set some notation. Let . Recall the maps from last time, which can be used to get pullbacks on cohomology. So we look at . Now, the definition is simple, the Gromov-Witten invariant .
That’s all there is to the definition. Just the integral of a class on the moduli space we discussed last time. Of course, this is oversimplifying things. The devil is in the details. First up, these are almost all zero. In fact, if the are homogeneous classes, then the Gromov-Witten invariant is zero unless .
So now we take . IE, this is the set of maps with no automorphisms and irreducible domain.
Lemma: If , then is a dense, open subset of .
Proof: If , then unless . Then , and so the result is trivial.
For , we have that is dense and open. So it is enough to show that for and general, there are no automorphisms. Now, is finite, since . There exists an open subset such that if and only if . Choosing the finishes the proof. QED.
Now, let be a pure dimensional subvariety of representing the class , with codimensions satisfying the above equation. Set, for , to be the translate by of .
Lemma: Let , for general. Then the scheme-theoretic intersection is a finite number of reduced points supported in , and this number is equal to .
The first part of this lemma is (apparently) a quick consequence of the Kleiman-Bertini Theorem. However, I’ve not actually been able to identify exactly which theorem this is, and if anyone can tell me the precise statement of the theorem, I’d much appreciate it. To see that the two numbers are the same is just a short cohomological diagram chase, and I’ll leave that to the reader.
In light of this lemma, we can see that is the number of pointed maps representing the class such that , which connects the Gromov-Witten invariants with actual pieces of enumerative data! Now, without proof, we’ll give a couple of rules that the Gromov-Witten invariants satisfy.
- unless , then .
- If , then for , we have is the pullback along of some class, so the fiber is positive dimensional. Thus, if , then unless , then .
- If , , then .
And finally we note that the Gromov-Witten invariants are symmetric in the , as we’re only looking at even cohomology classes.
Well, that’s all for this post. Next time, we’ll talk a bit about Quantum-Cohomology, a nice way to organize Gromov-Witten invariants, which can be used to compute them in many cases.