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arXiv · 2210.13411

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

Zhiyu Liu·Yongbin Ruan
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Abstract

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 $\leq 5$, which may be of independent interest.

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Zhiyu Liu, Yongbin Ruan. 2022-10-24. Castelnuovo bound and higher genus Gromov-Witten invariants of quintic 3-folds. https://arxiv.org/abs/2210.13411

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