## Gromov-Witten Invariants

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.

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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### 3 Responses to Gromov-Witten Invariants

1. Steven Sam says:

I like your posts a lot, it’s very helpful to see algebraic geometry explained in a less technical way than textbooks. Keep up the good work!

By the way, I think the Kleiman-Bertini theorem you are looking for is Theorem III.10.8 of Hartshorne. I’ll repeat it here for convenience:

Let X be a homogeneous space for G over an alg. closed field k of char. 0. Let Y and Z be nonsingular varieties with maps f: Y->X and g: Z->X. For g \in G, let gY be the composition $Y \xrightarrow{f} X \xrightarrow{g} X$. Then there is some Zariski open U in G such that for g \in U, the fiber product (in our case intersection) $gY \times_X Z$ is nonsingular and either empty or of dimension dim Y + dim Z – dim X.

• Steven Sam says:

Er, I used g twice for group elements and the map from Z to X, but I can’t seem to edit my post.