The multiple cover formula for reduced Gromov–Witten invariants of symplectic surfaces

Let $S$ be a symplectic surface, and let $\beta\in H_2(S,\mathbb{Z})$ be an effective curve class. For every positive integer $k$ dividing $\beta$, let

$$\varphi_k:H^{\ast}(S,\mathbb{Q})\to H^{\ast}(S_k,\mathbb{Q})$$

be a degree-preserving $\mathbb{Q}$-algebra isomorphism, where $S_k$ is a symplectic surface of the same type as $S$, satisfying $\varphi_k(\mathsf{p})=\mathsf{p}$, with $\mathsf{p}\in H^4(S,\mathbb{Q})$ the class of a point, and such that $\varphi_k(\beta/k)$ is an effective primitive curve class. Multiple Cover Formula. Then

$$\left\langle \tau_{a_1}(\gamma_1)\cdots\tau_{a_n}(\gamma_n)\right\rangle^S_{g,\beta}=\sum_{k\mid\beta}k^{2g-3+\sum_{i=1}^n\deg_{\mathbb{C}}(\gamma_i)}\left\langle \tau_{a_1}(\varphi_k(\gamma_1))\cdots\tau_{a_n}(\varphi_k(\gamma_n))\right\rangle^{S_k}_{g,\varphi_k(\beta/k)}.$$

If such a $\varphi_k$ does not exist, the corresponding summand is defined to vanish. This conjecture is the basic structural property expected for reduced Gromov–Witten theory of $K3$ and abelian surfaces; it expresses all invariants in terms of primitive classes, but its general status is not established in the supplied text.