Gromov–Witten 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 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 $$\langle \gamma_1, \dots, \gamma_n \rangle_{g, \beta} = \int_{[\overline{M}_{g,n}(X, \beta)]^{\mathrm{vir}}} \mathrm{ev}_1^* \gamma_1 \cup \cdots \cup \mathrm{ev}_n^* \gamma_n \in \Q.$$