Curves on K3 surfaces and modular forms
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1001.2719.pdf
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Author(s)
Pandharipande, R.
Thomas, R. P.
Maulik, Davesh
Date Issued
December 2016
Journal
Journal of Topology
Publisher
Oxford University Press (OUP)
Citation
Maulik, D., R. Pandharipande, and R. P. Thomas. “Curves on K
3 Surfaces and Modular Forms.” Journal of Topology 3, no. 4 (2010): 937–996.
Version
Author's final manuscript
Abstract
We study the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes. Equivalences relating the reduced Gromov-Witten invariants of K3surfaces to characteristic numbers of stable pairs moduli spaces are proved. As a consequence, we prove the Katz-Klemm-Vafa conjecture evaluating λ g integrals (in all genera) in terms of explicit modular forms. Indeed, all K3 invariants in primitive classes are shown to be governed by modular forms. The method of proof is by degeneration to elliptically fibred rational surfaces. New formulas relating reduced virtual classes on K3 surfaces to standard virtual classes after degeneration are needed for both maps and sheaves. We also prove a Gromov-Witten/Pairs correspondence for toric 3-folds. Our approach uses a result of Kiem and Li to produce reduced classes. In Appendix A, we answer a number of questions about the relationship between the Kiem-Li approach, traditional virtual cycles, and symmetric obstruction theories. The interplay between the boundary geometry of the moduli spaces of curves, K3 surfaces, and modular forms is explored in Appendix B by Pixton.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
http://dx.doi.org/10.1112/jtopol/jtq030
