The Gopakumar–Vafa finiteness conjecture

Let $(X,\omega)$ be a closed symplectic $6$-manifold. For every $A\in H_2(X,\mathbf Z)$ with $A\neq 0$ and $c_1(A)=0$, let $\mathrm{BPS}_{A,g}(X,\omega)$ be the numbers defined by the Gopakumar–Vafa formula relating them to the Gromov–Witten invariants. The Gopakumar–Vafa conjecture. These numbers satisfy

$$\mathrm{BPS}_{A,g}(X,\omega)\in\mathbf Z\quad\text{for every }g\in\mathbf N_0,$$

and there is a $g_A\in\mathbf N_0$ such that

$$\mathrm{BPS}_{A,g}(X,\omega)=0\quad\text{for every }g\geqslant g_A.$$

The conjecture asserts integrality and finiteness of the BPS invariants predicted by Gopakumar and Vafa. The integrality part has been proved by Ionel, while the finiteness assertion remains the part represented by the finiteness conjecture.