Marked relative invariants and GW/PT correspondences

Abstract

We introduce marked relative Pandharipande-Thomas (PT) invariants for a pair $(X,D)$ of a smooth projective threefold and a smooth divisor. These invariants are defined by integration over the moduli space of $r$-marked stable pairs on $(X,D)$, and appear naturally when degenerating diagonal insertions via the Li-Wu degeneration formula. We propose a Gromov-Witten (GW) / PT correspondence for marked relative invariants. We show compatibility of the conjecture with the degeneration formula and a splitting formula for relative diagonals. The results provide new tools to prove GW/PT correspondences for varieties with vanishing cohomology. As an application we prove the GW/PT correspondence for: (i) all Fano complete intersections, and (ii) the reduced theories of $(S \times C, S \times \{ z_1, \ldots, z_N \})$ where $S$ is a K3 surface and $C$ is a curve, for all curve classes which have divisibility at most $2$ over the K3 surface. In the appendix we introduce a notion of higher-descendent invariants which can be seen as an analogue of the nodal Gromov-Witten invariants defined by Arg\"uz, Bousseau, Pandharipande and Zvonkine in \cite{ABPZ}. We show that the higher-descendent invariants reduce to marked relative invariants with diagonal insertions.