Notes on the proof of the KKV conjecture

TL;DR

This paper surveys the proof of the KKV conjecture, which relates Gromov-Witten invariants of K3 surfaces to modular forms, marking a significant advance in understanding non-toric geometries in higher dimensions.

Contribution

It provides a comprehensive overview of the proof of the KKV conjecture, including new correspondences and computational techniques for Gromov-Witten theory of K3 surfaces.

Findings

Proof of the KKV conjecture established the Gromov-Witten theory of K3 surfaces in all genera.

Development of a new Pairs/Noether-Lefschetz correspondence for K3-fibred 3-folds.

Application of degeneration, localisation, and multiple cover formulae in calculations.

Abstract

The Katz-Klemm-Vafa conjecture expresses the Gromov-Witten theory of K3 surfaces (and K3-fibred 3-folds in fibre classes) in terms of modular forms. Its recent proof gives the first non-toric geometry in dimension greater than 1 where Gromov-Witten theory is exactly solved in all genera. We survey the various steps in the proof. The MNOP correspondence and a new Pairs/Noether-Lefschetz correspondence for K3-fibred 3-folds transform the Gromov-Witten problem into a calculation of the full stable pairs theory of a local K3-fibred 3-fold. The stable pairs calculation is then carried out via degeneration, localisation, vanishing results, and new multiple cover formulae.