Marked relative invariants and GW/PT correspondences.(English)Zbl 1544.14054
This paper proves the conjectural Gromov-Witten/Pandharipande-Thomas (GW/PT) correspondence, in the form stated byR. Pandharipande andA. Pixton [Geom. Topol. 18, No. 5, 2747–2821 (2014;Zbl 1342.14114)], for Fano complete intersections with arbitrary (not necessarily of even degree) cohomology insertions, and for the relative reduced theories of \(K3\times \text{curve}\) with some restrictions on the curve class. The former case is done inductively by a simple degeneration to lower-degree complete intersections. The latter case is done by reduction to the case of capped descendent invariants of \(K3\times \mathbb{P}^1\) relative to \(K3\times \{\infty\}\), monodromy arguments (of the \(K3\) surface), and finally a simple degeneration to two rational elliptic surfaces glued along an elliptic curve. The relevant GW/PT correspondences in the resulting base cases were proved in [R. Pandharipande andA. Pixton, J. Am. Math. Soc. 30, No. 2, 389–449 (2017;Zbl 1360.14134)].
For these arguments to work, the author generalizes the notion of relative PT stable pairs [J. Li andB. Wu, Commun. Anal. Geom. 23, No. 4, 841–921 (2015;Zbl 1349.14014)] to allow for markings, and provides detailed and fairly self-contained proofs (with references when they are identical to existing work) of its fundamental properties, in particular that marked relative PT invariants satisfy the same degeneration and splitting formulas as marked relative GW invariants. A generalization of the conjectural GW/PT correspondence to marked invariants, including an explicit descendent transformation, is stated and proved to be compatible with degeneration and splitting, and, in the case of \(K3\times \text{curve}\), multiple cover formulas.
For these arguments to work, the author generalizes the notion of relative PT stable pairs [J. Li andB. Wu, Commun. Anal. Geom. 23, No. 4, 841–921 (2015;Zbl 1349.14014)] to allow for markings, and provides detailed and fairly self-contained proofs (with references when they are identical to existing work) of its fundamental properties, in particular that marked relative PT invariants satisfy the same degeneration and splitting formulas as marked relative GW invariants. A generalization of the conjectural GW/PT correspondence to marked invariants, including an explicit descendent transformation, is stated and proved to be compatible with degeneration and splitting, and, in the case of \(K3\times \text{curve}\), multiple cover formulas.
Reviewer: Huaxin Liu (Oxford)
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 14J28 | \(K3\) surfaces and Enriques surfaces |
Keywords:
Gromov-Witten theory;Pandharipande-Thomas theory;GW/PT correspondence;\(K3\) surfaces;Fano hypersurfacesCite
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