THE KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACES
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/P...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2016-01-01
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| Series: | Forum of Mathematics, Pi |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050508616000020/type/journal_article |
| _version_ | 1811156257189396480 |
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| author | R. PANDHARIPANDE R. P. THOMAS |
| author_facet | R. PANDHARIPANDE R. P. THOMAS |
| author_sort | R. PANDHARIPANDE |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-10T04:48:37Z |
| format | Article |
| id | doaj.art-c1eea11ca7fc4866ae560cf4b2d16ecd |
| institution | Directory Open Access Journal |
| issn | 2050-5086 |
| language | English |
| last_indexed | 2024-04-10T04:48:37Z |
| publishDate | 2016-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Pi |
| spelling | doaj.art-c1eea11ca7fc4866ae560cf4b2d16ecd2023-03-09T12:34:23ZengCambridge University PressForum of Mathematics, Pi2050-50862016-01-01410.1017/fmp.2016.2THE KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACESR. PANDHARIPANDE0R. P. THOMAS1Departement Mathematik, ETH Zürich, Switzerland;Department of Mathematics, Imperial College, UK;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.https://www.cambridge.org/core/product/identifier/S2050508616000020/type/journal_article14N35 |
| spellingShingle | R. PANDHARIPANDE R. P. THOMAS THE KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACES Forum of Mathematics, Pi 14N35 |
| title | THE KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACES |
| title_full | THE KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACES |
| title_fullStr | THE KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACES |
| title_full_unstemmed | THE KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACES |
| title_short | THE KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACES |
| title_sort | katz klemm vafa conjecture for k3 surfaces |
| topic | 14N35 |
| url | https://www.cambridge.org/core/product/identifier/S2050508616000020/type/journal_article |
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