2021/12/22 by Georg Oberdieck, Oberdieck, Georg · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2112.11949
openalex publication_date 2021/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeABPZ. We show that the higher-descendent invariants reduce to marked relative invariants with diagonal insertions.