Curve counting on the Enriques surface and the Klemm-Mariño formula
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909410941468672 |
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| author | Oberdieck, Georg |
| author_facet | Oberdieck, Georg |
| contents | We determine the Gromov-Witten invariants of the local Enriques surfaces for all genera and curve classes and prove the Klemm-Mariño formula. In particular, we show that the generating series of genus $1$ invariants of the Enriques surface is the Fourier expansion of a certain power of Borcherds automorphic form on the moduli space of Enriques surfaces. We also determine all Vafa-Witten invariants of the Enriques surface.
The proof uses the correspondence between Gromov-Witten and Pandharipande-Thomas theory. On the Gromov-Witten side we prove the relative Gromov-Witten potentials of an elliptic Enriques surfaces are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. On the sheaf side, we relate the Pandharipande-Thomas invariants of the Enriques-Calabi-Yau threefold in fiber classes to the $2$-dimensional Donaldson-Thomas invariants by a version of Toda's formula for local K3 surfaces. Altogether, we obtain sufficient modular constraints to determine all invariants from basic geometric computations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_11115 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Curve counting on the Enriques surface and the Klemm-Mariño formula Oberdieck, Georg Algebraic Geometry We determine the Gromov-Witten invariants of the local Enriques surfaces for all genera and curve classes and prove the Klemm-Mariño formula. In particular, we show that the generating series of genus $1$ invariants of the Enriques surface is the Fourier expansion of a certain power of Borcherds automorphic form on the moduli space of Enriques surfaces. We also determine all Vafa-Witten invariants of the Enriques surface. The proof uses the correspondence between Gromov-Witten and Pandharipande-Thomas theory. On the Gromov-Witten side we prove the relative Gromov-Witten potentials of an elliptic Enriques surfaces are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. On the sheaf side, we relate the Pandharipande-Thomas invariants of the Enriques-Calabi-Yau threefold in fiber classes to the $2$-dimensional Donaldson-Thomas invariants by a version of Toda's formula for local K3 surfaces. Altogether, we obtain sufficient modular constraints to determine all invariants from basic geometric computations. |
| title | Curve counting on the Enriques surface and the Klemm-Mariño formula |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2305.11115 |