Surveys in Differential Geometry
Volume 21 (2016)
Notes on the proof of the KKV conjecture
Pages: 289 – 311
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.