---
tag: 9YTV2296X7
kind: definition
name: Gromov–Witten invariant
msc: [14N35, 53D45]
defines: XVJA0JD79P
uses: [FJZRW9P0K3, VNX79CRH90, RDT4785M4D]
sources:
  - url: https://en.wikipedia.org/wiki/Gromov%E2%80%93Witten_invariant
---

## Definition

Let $X$ be a smooth projective variety over $\C$ of dimension $r$.
Let $g, n \ge 0$ be integers and let $\beta \in H_2(X, \Z)$.
A *stable map* of genus $g$ with $n$ marked points and class $\beta$ is a tuple $(C, x_1, \dots, x_n, f)$ such that:

1. $C$ is a connected projective curve of arithmetic genus $g$ whose only singularities are nodes.
2. $x_1, \dots, x_n$ are distinct smooth points of $C$.
3. $f \colon C \to X$ is a morphism with $f_*[C] = \beta$.
4. The tuple has finitely many automorphisms.

Let $\overline{M}_{g,n}(X, \beta)$ be the [[moduli stack|FJZRW9P0K3]] of such stable maps, with evaluation maps $\mathrm{ev}_i \colon \overline{M}_{g,n}(X, \beta) \to X$, $\mathrm{ev}_i(C, x, f) = f(x_i)$.
It carries a virtual fundamental class $[\overline{M}_{g,n}(X, \beta)]^{\mathrm{vir}}$ of degree
$$(1 - g)(r - 3) + \int_\beta c_1(T_X) + n.$$
For $\gamma_1, \dots, \gamma_n \in H^*(X, \Q)$, the *Gromov–Witten invariant* is
$$\langle \gamma_1, \dots, \gamma_n \rangle_{g, \beta} = \int_{[\overline{M}_{g,n}(X, \beta)]^{\mathrm{vir}}} \mathrm{ev}_1^* \gamma_1 \cup \cdots \cup \mathrm{ev}_n^* \gamma_n \in \Q.$$

## Remarks

Example: let $X = \mathbb{P}^2$, $g = 0$, $\beta = d$ times the class of a line, and $n = 3d - 1$.
The virtual dimension is $-1 + 3d + (3d - 1) = 6d - 2$, and each point class has complex degree $2$, so $3d - 1$ point classes fill the dimension exactly.
The invariant $N_d = \langle \mathrm{pt}, \dots, \mathrm{pt} \rangle_{0, d}$ is the number of rational plane curves of degree $d$ through $3d - 1$ general points, with $N_1 = 1$, $N_2 = 1$ and $N_3 = 12$.

Non-example: let $C$ have a genus $0$ component that $f$ contracts to a point.
If that component carries only two special points, nodes or marked points, the map is not stable.
That component has infinitely many automorphisms fixing its two special points, so clause 4 fails.

The same construction works for a compact symplectic manifold with a compatible almost complex structure, with [[pseudoholomorphic curves|N469E47V75]] in place of algebraic maps.
The invariant is zero unless the degrees of the $\gamma_i$ add up to the virtual dimension.