The multiple cover formula for $K3$ and abelian surfaces

Abstract

All reduced descendent Gromov-Witten invariants of $K3$ and abelian surfaces in primitive curve classes can be calculated by the methods of \cite{BOPY,MPT}. To handle the imprimitive curve classes, a multiple cover formula was conjectured in \cite{ObPand} for $K3$ surfaces and in \cite{O_NLGW} for abelian surfaces. We prove here that both descendent multiple cover formulas are implied by the conjectural families GW/PT correspondence for semipositive relative 3-folds with primary insertions. The implication is proven by showing that the multiple cover formula for $S$ can be recast as a property of an appropriate localization vertex for the relative 3-fold Gromov-Witten theory of $(S\times \mathbb{P}^1/S_0 \cup S_\infty)$. The families GW/PT correspondence then transfers the multiple cover formula from the Gromov-Witten side to the stable pairs side where the formula is proven geometrically by studying cosections and applying universality properties. Along the way, we prove a DT/PT correspondence for the reduced theories of $(S\times \mathbb{P}^1/S_0 \cup S_\infty)$ using the wallcrossing techniques of Kuhn-Liu-Thimm \cite{KLT2,KLT}.