THE KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACES. (6th June 2016)

• Record Type: Journal Article
• Authors: PANDHARIPANDE, R.; THOMAS, R. P.
• Is Part Of: Forum of mathematics. Volume 4(2016)
• Publication Date: 2016-06-06
• Subjects: 14N35; Mathematics -- Periodicals; 510
• DOI: 10.1017/fmp.2016.2
• Languages: English
• ISSNs: 2050-5086

Abstract

Abstract : We prove the KKV conjecture expressing Gromov–Witten invariants of $K3$ surfaces in terms of modular forms. Our results apply in every genus and for every curve class. The proof uses the Gromov–Witten/Pairs correspondence for $K3$ -fibered hypersurfaces of dimension 3 to reduce the KKV conjecture to statements about stable pairs on (thickenings of) $K3$ surfaces. Using degeneration arguments and new multiple cover results for stable pairs, we reduce the KKV conjecture further to the known primitive cases. Our results yield a new proof of the full Yau–Zaslow formula, establish new Gromov–Witten multiple cover formulas, and express the fiberwise Gromov–Witten partition functions of $K3$ -fibered 3-folds in terms of explicit modular forms.