Surveys in Differential Geometry

Volume 21 (2016)

Notes on the proof of the KKV conjecture

Pages: 289 – 311

DOI: http://dx.doi.org/10.4310/SDG.2016.v21.n1.a7

Authors

Rahul Pandharipande (Departement Mathematik, ETH Zürich, Switzerland)

Richard P. Thomas (Department of Mathematics, Imperial College London, United Kingdom)

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.

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Published 7 June 2016