An open GW-formula for Lagrangians in Fano varieties

Given a Fano variety $Y$ and a simple normal crossings divisor $D\subseteq Y$ which is anti-canonical, we prove a formula relating counts of discs with boundary on a Lagrangian $L\subseteq Y\mathrm{\backslash }D$ to counts of rational curves in $Y$, under suitable positivity assumptions on $L$. This formula seriously constrains the topology of $L$ in many examples. Our main application is a super-potential formula for Fano cyclic coverings $X$ of $Y$. As a corollary, we show that all the small components of the Fukaya category of a Fano hypersurface $X\subseteq {\mathbb{P}}^{n+1}$ are split-generated by monotone Lagrangian tori.