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  "body": "\n## Definition\n\nLet $X$ be a smooth projective variety over $\\C$ of dimension $r$.\nLet $g, n \\ge 0$ be integers and let $\\beta \\in H_2(X, \\Z)$.\nA *stable map* of genus $g$ with $n$ marked points and class $\\beta$ is a tuple $(C, x_1, \\dots, x_n, f)$ such that:\n\n1. $C$ is a connected projective curve of arithmetic genus $g$ whose only singularities are nodes.\n2. $x_1, \\dots, x_n$ are distinct smooth points of $C$.\n3. $f \\colon C \\to X$ is a morphism with $f_*[C] = \\beta$.\n4. The tuple has finitely many automorphisms.\n\nLet $\\overline{M}_{g,n}(X, \\beta)$ be the [[moduli stack|FJZRW9P0K3]] of such stable maps, with evaluation maps $\\mathrm{ev}_i \\colon \\overline{M}_{g,n}(X, \\beta) \\to X$, $\\mathrm{ev}_i(C, x, f) = f(x_i)$.\nIt carries a virtual fundamental class $[\\overline{M}_{g,n}(X, \\beta)]^{\\mathrm{vir}}$ of degree\n$$(1 - g)(r - 3) + \\int_\\beta c_1(T_X) + n.$$\nFor $\\gamma_1, \\dots, \\gamma_n \\in H^*(X, \\Q)$, the *Gromov–Witten invariant* is\n$$\\langle \\gamma_1, \\dots, \\gamma_n \\rangle_{g, \\beta} = \\int_{[\\overline{M}_{g,n}(X, \\beta)]^{\\mathrm{vir}}} \\mathrm{ev}_1^* \\gamma_1 \\cup \\cdots \\cup \\mathrm{ev}_n^* \\gamma_n \\in \\Q.$$\n\n## Remarks\n\nExample: let $X = \\mathbb{P}^2$, $g = 0$, $\\beta = d$ times the class of a line, and $n = 3d - 1$.\nThe 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.\nThe 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 $N_3 = 12$.\n\nNon-example: let $C$ have a genus $0$ component that $f$ contracts to a point.\nIf that component carries only two special points, nodes or marked points, the map is not stable.\nThat component has infinitely many automorphisms fixing its two special points, so clause 4 fails.\n\nThe same construction works for a compact symplectic manifold with a compatible almost complex structure, with [[pseudoholomorphic curves|N469E47V75]] in place of algebraic maps.\nThe invariant is zero unless the degrees of the $\\gamma_i$ add up to the virtual dimension.\n",
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