The genus of a curve on Calabi-Yau threefolds

Abstract

Let $X$ be a smooth projective Calabi-Yau variety of dimension $3$, with Picard number one. Let $n$ be the self-intersection of the ample generator of the Néron-Severi group of $X$. Let $\mathsf{GV}_{g,d}$ be the Gopakumar-Vafa invariant of $X$ of genus $g$ and degree $d$. It is conjectured that {\it if $g>\frac{d^{2}}{2n}+\frac{d}{2}+1$, then $\mathsf{GV}_{g,d}=0$}. In this paper, we prove the implication when $d\gg n$ and $X\subset \mathbb P^r$ is a complete intersection in projective space. The vanishing follows by proving that if $C$ is a closed subscheme of $X$ of dimension $1$, arithmetic genus $p_a(C)$, and degree $d\gg n$, then $p_a(C) \leq \frac{d^2}{2n}+\frac{d}{2}+1$. Actually, we prove a somewhat more general result. In particular, we prove that $C$ is an extremal curve if and only if $C$ is the complete intersection on $X$ with a hyperplane $H$ and a hypersurface $F$, namely $C=X\cap H\cap F$.