Universally counting curves in Calabi--Yau threefolds

TL;DR

This paper proves that curve counting invariants for complex threefolds with nef anti-canonical bundle are determined by local curve data, confirming the MNOP conjecture for a broad class of threefolds including Calabi--Yau threefolds.

Contribution

It establishes the MNOP conjecture for all complex threefolds with nef anti-canonical bundle by reducing the problem to local curves and introduces a new transversality result for holomorphic curves.

Findings

MNOP conjecture holds for all threefolds with nef anti-canonical bundle

Curve invariants are determined by local curve data

New transversality result for holomorphic curves

Abstract

We show that curve enumeration invariants of complex threefolds with nef anti-canonical bundle are determined by their values on local curves. This implies the MNOP conjecture of Maulik, Nekrasov, Okounkov, and Pandharipande relating Gromov--Witten and Donaldson--Pandharipande--Thomas invariants, for all complex threefolds with nef anti-canonical bundle (in particular, all Calabi--Yau threefolds) and primary insertions (no descendents), given its known validity for local curves due to Bryan, Okounkov, and Pandharipande. The main new technical ingredient in our work is a generic transversality result for holomorphic curves in complex manifolds. Due to the rigidity of complex structures, this result is necessarily weaker than the corresponding generic transversality property for holomorphic curves in almost complex manifolds. Despite this weaker nature, it is enough to obtain our main result by following the proof of the Gopakumar--Vafa integrality conjecture by Ionel and Parker.