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×C,S×{z1,…,zN}) 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 Argüz, Bousseau, Pandharipande and Zvonkine in [3]. We show that the higher-descendent invariants reduce to marked relative invariants with diagonal insertions.