The Katz-Klemm-Vafa conjecture for K3 surfaces
File(s)
Published version
Author(s)
Pandharipande, R
Thomas, RP
Type
Journal Article
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.
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.
Date Issued
2016-06-06
Date Acceptance
2016-04-26
Citation
Forum of Mathematics, Pi, 2016, 4
ISSN
2050-5086
Publisher
Cambridge University Press
Journal / Book Title
Forum of Mathematics, Pi
Volume
4
Copyright Statement
© The Author(s) 2016 This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
License URL
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/G06170X/1
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
RATIONAL CURVES
REDUCED CLASSES
STABLE PAIRS
FORMULA
math.AG
math.AG
hep-th
math.SG
14N35
Publication Status
Published
Article Number
e4
Date Publish Online
2016-06-06