Curves on K3 surfaces in divisibility 2
We prove a conjecture of Maulik, Pandharipande and Thomas expressing the Gromov–Witten invariants of K3 surfaces for divisibility 2 curve classes in all genera in terms of weakly holomorphic quasi-modular forms of level 2. Then we establish the holomorphic anomaly equation in divisibility 2 in all g...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2021-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509421000062/type/journal_article |
| _version_ | 1811156195678879744 |
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| author | Younghan Bae Tim-Henrik Buelles |
| author_facet | Younghan Bae Tim-Henrik Buelles |
| author_sort | Younghan Bae |
| collection | DOAJ |
| description | We prove a conjecture of Maulik, Pandharipande and Thomas expressing the Gromov–Witten invariants of K3 surfaces for divisibility 2 curve classes in all genera in terms of weakly holomorphic quasi-modular forms of level 2. Then we establish the holomorphic anomaly equation in divisibility 2 in all genera. Our approach involves a refined boundary induction, relying on the top tautological group of the moduli space of smooth curves, together with a degeneration formula for the reduced virtual fundamental class with imprimitive curve classes. We use double ramification relations with target variety as a new tool to prove the initial condition. The relationship between the holomorphic anomaly equation for higher divisibility and the conjectural multiple cover formula of Oberdieck and Pandharipande is discussed in detail and illustrated with several examples. |
| first_indexed | 2024-04-10T04:47:27Z |
| format | Article |
| id | doaj.art-b8e0d926e2394502bf86d5f6c7b49f26 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:27Z |
| publishDate | 2021-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-b8e0d926e2394502bf86d5f6c7b49f262023-03-09T12:34:52ZengCambridge University PressForum of Mathematics, Sigma2050-50942021-01-01910.1017/fms.2021.6Curves on K3 surfaces in divisibility 2Younghan Bae0Tim-Henrik Buelles1ETH Zürich, Department of Mathematics, Zürich, Switzerland; E-mail:ETH Zürich, Department of Mathematics, Zürich, Switzerland; E-mail:We prove a conjecture of Maulik, Pandharipande and Thomas expressing the Gromov–Witten invariants of K3 surfaces for divisibility 2 curve classes in all genera in terms of weakly holomorphic quasi-modular forms of level 2. Then we establish the holomorphic anomaly equation in divisibility 2 in all genera. Our approach involves a refined boundary induction, relying on the top tautological group of the moduli space of smooth curves, together with a degeneration formula for the reduced virtual fundamental class with imprimitive curve classes. We use double ramification relations with target variety as a new tool to prove the initial condition. The relationship between the holomorphic anomaly equation for higher divisibility and the conjectural multiple cover formula of Oberdieck and Pandharipande is discussed in detail and illustrated with several examples.https://www.cambridge.org/core/product/identifier/S2050509421000062/type/journal_article14N3514J2811F03 |
| spellingShingle | Younghan Bae Tim-Henrik Buelles Curves on K3 surfaces in divisibility 2 Forum of Mathematics, Sigma 14N35 14J28 11F03 |
| title | Curves on K3 surfaces in divisibility 2 |
| title_full | Curves on K3 surfaces in divisibility 2 |
| title_fullStr | Curves on K3 surfaces in divisibility 2 |
| title_full_unstemmed | Curves on K3 surfaces in divisibility 2 |
| title_short | Curves on K3 surfaces in divisibility 2 |
| title_sort | curves on k3 surfaces in divisibility 2 |
| topic | 14N35 14J28 11F03 |
| url | https://www.cambridge.org/core/product/identifier/S2050509421000062/type/journal_article |
| work_keys_str_mv | AT younghanbae curvesonk3surfacesindivisibility2 AT timhenrikbuelles curvesonk3surfacesindivisibility2 |