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 X 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 \mathrm{Aut}(X) modulo the stabilizer of a point). These are generally nice spaces. For one thing, they’re always convex. Among the examples are algebraic groups, \mathbb{P}^n, 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 A_i(X)=H_{2i}(X,\mathbb{Z}) and A^i(X)=H^{2i}(X,\mathbb{Z}), 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 \gamma_1,\ldots,\gamma_n\in A^*(X). Recall the maps ev_i from last time, which can be used to get pullbacks on cohomology. So we look at (ev_1^*\gamma_1)\cup\ldots\cup (ev_n^*\gamma_n)\in H^*(\bar{M}_{0,n}(X,\beta)). Now, the definition is simple, the Gromov-Witten invariant I_\beta(\gamma_1,\ldots,\gamma_n)=\int_{\bar{M}_{0,n}(X,\beta)} (ev_1^*\gamma_1)\cup\ldots\cup(ev_n^*\gamma_n).

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 \gamma_i are homogeneous classes, then the Gromov-Witten invariant is zero unless \sum_{i=1}^n \mathrm{codim}(\gamma_i)=\dim X+\int_\beta c_1(T_X)+n-3=\dim \bar{M}_{0,n}(X,\beta).
So now we take M^*_{0,n}(X,\beta)=M_{0,n}(X,\beta)\cap \bar{M}^*_{0,n}(X,\beta). IE, this is the set of maps with no automorphisms and irreducible domain.

Lemma: If n\geq 1, then M^*_{0,n}(X,\beta) is a dense, open subset of \bar{M}_{0,n}(X,\beta).

Proof: If \beta=0, then \bar{M}_{0,n}(X,0)=\emptyset unless n\geq 3. Then \bar{M}^*_{0,n}(X,0)=\bar{M}_{0,n}(X,\beta), and so the result is trivial.

For \beta\neq 0, we have that M_{0,n}(X,\beta)\subseteq \bar{M}_{0,n}(X,\beta) is dense and open. So it is enough to show that for \mathbb{P}^1 and p_i general, there are no automorphisms. Now, \mathrm{Aut}(\mu:\mathbb{P}^1\to X) is finite, since \beta\neq 0. There exists an open subset U\subset \mathbb{P}^1 such that p\in U if and only if stab(p)=\{e\}. Choosing the p_i\in U finishes the proof. QED.

Now, let \Gamma_i be a pure dimensional subvariety of X representing the class \gamma_i, with codimensions satisfying the above equation. Set, for g\in \mathrm{Aut}(X), g\Gamma_i to be the translate by g of \Gamma_i.

Lemma: Let n\geq 0, for g_1,\ldots,g_n\in\mathrm{Aut}(X) general. Then the scheme-theoretic intersection \bigcap_{i=1}^n ev_i^{-1}(g_i\Gamma_i) is a finite number of reduced points supported in M_{0,n}(X,\beta), and this number is equal to I_\beta(\gamma_1,\ldots,\gamma_n).

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 I_\beta(\gamma_1,\ldots,\gamma_n) is the number of pointed maps \mu:\mathbb{P}^1\to X representing the class \beta such that \mu(p_i)\in g_i\Gamma_i, 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.

  1. I_0(\gamma_1,\ldots,\gamma_n)=0 unless n=3, then I_0(\gamma_1,\gamma_2,\gamma_3)=\int_X \gamma_1\cup\gamma_2\cup\gamma_3.
  2. If \gamma_1=1\in A^0X, then for \beta\neq 0, we have ev_1^*(\gamma_1)\cup\ldots \cup ev_n^*(\gamma_n) is the pullback along \bar{M}_{0,n}(X,\beta)\to \bar{M}_{0,n-1}(X,\beta) of some class, so the fiber is positive dimensional. Thus, if \gamma_1=1, then I_\beta(\gamma_1,\ldots,\gamma_n)=0 unless n=3,\beta=0, then I_0(1,\gamma_2,\gamma_3)=\int_X \gamma_2\cup\gamma_3.
  3. If \gamma_1\in A^1X, \beta\neq 0, then I_\beta(\gamma_1,\ldots,\gamma_n)=\left(\int_\beta \gamma_1\right)I_\beta(\gamma_2,\ldots,\gamma_n).

And finally we note that the Gromov-Witten invariants are symmetric in the \gamma_i, 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.

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