Castelnuovo bound and higher genus Gromov-Witten invariants of quintic 3-folds

We prove a conjectural vanishing result for Gopakumar–Vafa invariants of quintic 3-folds, referred to as Castelnuovo bound in the literature. Furthermore, we calculate Gopakumar–Vafa invariants at Castelnuovo bound $g=\frac{d^2+5d+10}{10}$. As physicists showed, these two properties allow us to compute all Gromov–Witten invariants of quintic 3-folds up to genus $53$, provided that the conifold gap condition holds. We also give a bound for the genus of any one-dimensional closed subscheme in a smooth hypersurface of degree $\le 5$, which may be of independent interest.