Gromov-Witten Invariants Notes

Fix a projective algebraic variety \(X\) and fix a homology class \(\beta\in H_2(X,\mathbb Z)\) and cycles \(Z_1,...,Z_n\) (I’m thinking of these as divisors). The basic question concerns the structure of the following set of curves:

\begin{equation} \label{eq:1} C\subset X ~ \text{of genus $g$, homology class $\beta$, and $C\cap Z_{i}\neq \emptyset$ for all $i$} \end{equation}

assuming that the \(Z_i\) are in general position. Kontesevich made the key observation that curves \(C\subset X\) should be replaced with $n$-pointed curves \((C,p_1,...,p_n)\) and holomorphic maps \(f:C\to X\). This observation is apparently consistent with the “origins of the notion in the nonlinear sigma model and two dimensional TQFT coupled to gravity” Chapter 7, page 168. Then

\begin{equation} \label{eq:3} f:C\to X~ \text{ such that }~ f_{*}[C] = \beta ~ \text{ and } f(p_{i}) \in Z_{i} ~ \text{ for } ~ i = 1,...,n \end{equation}

There is an obvious notion of isomorphism of such maps (an isomorphim of pointed curves \(\varphi: C\cong C'\) such that \(f = f'\circ \varphi\)) so we need to consider their moduli. To get a compact moduli space yada yada we need reducible nodal curves of genus \(g\). Then the set above should cut out a closed subset of \(\overline{M}_{g,n}\). Intuitively, then, the Gromov-Witten class

\begin{equation} \label{eq:4} I_{g,n,\beta} (\alpha_1, ...,\alpha_{n}) \in H^{*}(\overline{M}_{g,n},\mathbb Q) \end{equation}

is supposed to be the cohomology class represented by this closed subset (here \(\alpha_i\) is the cohomology class dual to the homology class \(Z_i\)). This means we can interpret Gromov-Witten classes as a system of maps

\begin{equation} \label{eq:5} I_{g,n,\beta}:H^{*}(X,\mathbb Q)^{\otimes n} \to H^{*}(\overline M_{g,n},\mathbb Q). \end{equation}

The Gromov-Witten invariants are defined similarly; simply cap the Gromov-Witten class with the fundamental class of \(\overline M_{g,n}\):

\begin{equation} \label{eq:6} \langle I_{g,n,\beta}\rangle (\alpha_1,...,\alpha_n) = \int_{\overline M_{g,n}} I_{g,n,\beta} (\alpha_1, ...,\alpha_{n}). \end{equation}

Equation \ref{eq:6} above vanishes unless the Gromov-Witten class \(I_{g,n,\beta} (\alpha_1, ...,\alpha_{n})\) has some component in the top degree of \(H^*(\overline M_{g.n},\mathbb Q)\), which intuitively happen when the set \ref{eq:3} consists of only finitely many curves. However, even when this is the case, the GW-invariants will often be fractional or negative, making their enumerative significance…unclear.

This is all quite naive. To define Gromov-Witten classes we’ll need to study the moduli of maps above more carefully, and in particular will need stable maps and virtual fundamental classes.

Here we review stable maps.

stable-map-definition

Note that given the first and second axioms above, the third and fourth together are equivalent to requiring that \(C\) has finitely many automorphisms.

stable-map-family-definition

moduli-space-of-stable-maps

Let \(X\) be projective algebraic variety and fix a class \(\beta\in H_2(X,\mathbb Z)\). Then we can define a moduli functor (contravariant)

\begin{equation} \overline{\mathcal M}_{g,n}(X,\beta):(\mathbb C -\mathsf{Schemes})\to (\mathsf{Sets}) \end{equation}

by sending a scheme \(S\) to the set of all isomorphism classes of families of stable maps over \(S\) of genus \(g\) and class \(\beta\).

To solve this moduli problem one must represent this functor in some way. The straightforward approach is to show there exists a course moduli space and the more sophisticated approach is to show that this moduli functor is actually and algebraic stack.

#+begin_theorem If \(X\) is a projective variety over \(\mathbb C\) and \(\beta\) is a homology class in \(H_2(X,\mathbb Z)\), then the functor \(\overline{\mathcal M}_{g,n}(X,\beta)\) has a coarse moduli space \(\overline{M}_{g.n}(X,\beta)\) which is a projective scheme over \(\mathbb C\). #+end_thorem

A fairly explicit constuction can be found in [FP, /Notes on stable maps and quantum cohomology/].

The underlying algebraic space of \(\overline{\mathcal M}_{g,n}(X,\beta)\) is the course moduli space \(\overline{M}_{g,n}(X,\beta)\).

We get the following natural maps:

which are defined as follows. Given a stable map \(f:(C,p_1,...,p_n)\to X\), we get a well-defined tuple

\begin{equation} \label{eq:full-evaluation-map} \pi_1(C,p_1,...,p_n,f) = (f(p_1),...,f(p_n)) \in X^n \end{equation}

defining \(\pi_1\) set-theoretically. It shouldn’t be too unbelievable that this is indeed a morphism as well. I think of this as the “full evaluation map”, since composition with the projection onto the $i$th factor of \(X^n\) gives the map \(\operatorname{ev}_{i}\):

\begin{align*} \operatorname{ev}_{i}:\overline M_{g,n} \to X ~,~ \operatorname{ev}_{i}(C,p_1,...,p_n,f) = f(p_{i}). \end{align*}

As for \(\pi_2\), it should just be the map which forgets the stable map \(f\). However, \(C\) need not be a stable curve if \(f:C\to X\) is a stable map; for instance, we could have a component \(C_i\cong \mathbb P^1\) of \(C\) which isn’t contracted by \(f\), in which case it need not have 3 specal points. However, once we contract the non-stable components of \(C\) we do get a stable curve \(\widetilde C\) – provided that \(n + 2g \geq 3\) (so that \(\overline M_{g,n}\) exists).

We define \(\pi_2\) set-theoretically by sending \(C\) to \(\widetilde C\).

Assume \(n + 2g \geq 3\) (so that \(\overline M_{g,n}\) exists) and let

\begin{equation} \label{eq:proj-map-1} \pi_1:\overline M_{g,n}(X,\beta)\to X^{n} \end{equation} \begin{equation} \label{eq:proj-map-2} \pi_2:\overline M_{g,n}(X,\beta)\to \overline M_{g,n} \end{equation}

be the full evaluation map and the contraction-forgetful map respectively (see moduli-space-of-stable-maps). We can use these to define Gromov-Witten classes in the special case that \(X\) is smooth and \(\overline M_{g,n}(X,\beta)\) is an orbifold of dimension

\begin{equation} \label{eq:expected-dimension} (1-g)(\dim X - 3) - \int_{\beta}\omega_{X} + n \end{equation}

where \(\omega_X\) is the canonical class of \(X\). Note that this is the expected dimension of \(\overline M_{g,n}(X,\beta)\). Pullback and pushforward give us the following maps on cohomology and homology respectively:

\begin{equation} \label{eq:pullback1} \pi_{1}^{*}:H^{*}(X,\mathbb Q)^{\otimes n} \to H^{*}(\overline M_{g,n}(X,\beta),\mathbb Q) \end{equation} \begin{equation} \label{eq:pullback2} \pi_{2*}:H_{*}(\overline M_{g,n}(X,\beta),\mathbb Q) \to H_{*}(\overline M_{g,n}, \mathbb Q) \end{equation}

All spaces involved are orbifolds, for which Poincare-Duality holds, so the pushforward \(\pi_{2*}\) induces a Gysin map

\begin{equation} \label{eq:gysin-map} \pi_{2!}:H^{*}(\overline M_{g,n}(X,\beta), \mathbb Q) \to H^{2m+*}(\overline M_{g,n},\mathbb Q) \end{equation}

where \(m = (g-1)\dim X + \int_\beta \omega_X\). We can then define Gromov-Witten classes for \(\alpha_i\in H^*(X,\mathbb Q)\) by pulling back along \(\pi_1\) to \(\overline M_{g,n}(X,\beta)\) and then projecting to \(\overline{M}_{g,n}\) with \(\pi_{2!}\):

\begin{equation} \label{eq:gromov-witten-definition-1} I_{g,n,\beta}(\alpha_1,...,\alpha_n) = \pi_{2!}(\pi_1^{*}(\alpha_{1}\otimes ...\otimes \alpha_n)). \end{equation}

This indeed picks out a cohomology class in \(\overline M_{g,n}\) as we expected in § Motivation and Intuition. This gives us a Gromov-Witten invariant

\begin{equation} \label{eq:gw-invariant-def-1} \langle I_{g,n,\beta}\rangle(\alpha_{1},...,\alpha_n) = \int_{\overline M_{g,n}} I_{g,n,\beta}(\alpha_1,...,\alpha_n) \end{equation}

whenever \(I_{g,n,\beta}(\alpha_1,...,\alpha_n)\) has dimension \(2(3g - 3 + n)\), the real dimension of \(\overline M_{g,n}\). Unwinding the definition of the Gysin homomorphism gives us the equality

\begin{equation} \label{eq:gw-def-1prime} \langle I_{g,n,\beta}\rangle(\alpha_{1},...,\alpha_n) = \int_{\overline M_{g,n}(X,\beta)} \pi_{1}^{*}(\alpha_1\otimes...\otimes\alpha_n). \end{equation}

which is the definition often found in modern literature (in my limited experience).

This definition works perfectly well, again, when \(\overline M_{g,n}(X,\beta)\) is a smooth orbifold of the expected dimension. A good example of this is when \(X = \mathbb P^r\) and \(g = 0\).

This doesn’t work in general, however. When \(g\geq 1\) or even for most manifolds \(X\neq \mathbb P^r\), \(\overline M_{g,n}(X,\beta)\) has components whose dimension exceeds the expected dimension. To see why this is a problem, we examine the roll played by the fundamental class

\begin{equation} \label{eq:fundamental-class} \xi = [\overline M_{g,n}(X,\beta)] \in H_{*}(\overline M_{g,n}(X,\beta), \mathbb Q) \end{equation}

which corresponds to \(1\in H^*(\overline M_{g,n}(X,\beta),\mathbb Q)\) under Poincare duality. Writing

\begin{equation} \label{eq:2} \pi:\overline M_{g,n}(X,\beta) \to X^{n} \times \overline M_{g,n} \end{equation}

for the product of \(\pi_1\) and \(\pi_2\) and denoting by \(p_1,p_2\) the natural projections above so that \(\pi_i = p_i\circ \pi\), we can write the Gromov-Witten classes as

\begin{equation} \label{eq:gw-with-fundamental-class} I_{g,n,\beta}(\alpha_1,...,\alpha_n) = PD^{-1}p_{2*}\big(p^{*}_1(\alpha_1\otimes...\otimes \alpha_n) \cap \pi_{*}(\xi)\big) \end{equation}

whenever \(n + 2g \geq 3\). The Gromov-Witten invariants can likewise be written

\begin{equation} \langle I_{g,n,\beta}\rangle(\alpha_{1},...,\alpha_n) = \int_{\xi} \pi_{1}^{*}(\alpha_1\otimes...\otimes\alpha_n). \end{equation}

or if we let \(\operatorname{ev}_{i}\) denote the $i$th evaluation map \((C,p_1,...,p_n,f)\mapsto f(p_i)\), then

\begin{equation} \label{eq:gw-invariants-with-FC} \langle I_{g,n,\beta}\rangle(\alpha_{1},...,\alpha_n) = \int_{\xi}\operatorname{ev}_{1}^{*}(\alpha_1)\cup...\cup\operatorname{ev}_n^{*}(\alpha_n). \end{equation}

These definitions will make sense whenever the fundemental class \(\xi\) is well-behaved, i.e. is pure dimensional + other nice properties probably. In general it will need to be replaced with a virtual fundamental class.

normal-cone