Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911783167459328 |
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| author | Cao, Yalong Oberdieck, Georg Toda, Yukinobu |
| author_facet | Cao, Yalong Oberdieck, Georg Toda, Yukinobu |
| contents | Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds, Klemm and Pandharipande for Calabi-Yau 4-folds, Pandharipande and Zinger for Calabi-Yau 5-folds.
We conjecture that our invariants are integers and give a sheaf-theoretic interpretation in terms of reduced $4$-dimensional Donaldson-Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two $K3$ surfaces and for the cotangent bundle of $\mathbb{P}^2$. Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a $K3$ surface. This yields a conjectural formula for the number of isolated genus $2$ curves of minimal degree on a very general hyperkähler $4$-fold of $K3^{[2]}$-type. The formula may be viewed as a $4$-dimensional analogue of the classical Yau-Zaslow formula concerning counts of rational curves on $K3$ surfaces.
In the course of our computations, we also derive a new closed formula for the Fujiki constants of the Chern classes of tangent bundles of both Hilbert schemes of points on $K3$ surfaces and generalized Kummer varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_10878 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds Cao, Yalong Oberdieck, Georg Toda, Yukinobu Algebraic Geometry High Energy Physics - Theory Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds, Klemm and Pandharipande for Calabi-Yau 4-folds, Pandharipande and Zinger for Calabi-Yau 5-folds. We conjecture that our invariants are integers and give a sheaf-theoretic interpretation in terms of reduced $4$-dimensional Donaldson-Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two $K3$ surfaces and for the cotangent bundle of $\mathbb{P}^2$. Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a $K3$ surface. This yields a conjectural formula for the number of isolated genus $2$ curves of minimal degree on a very general hyperkähler $4$-fold of $K3^{[2]}$-type. The formula may be viewed as a $4$-dimensional analogue of the classical Yau-Zaslow formula concerning counts of rational curves on $K3$ surfaces. In the course of our computations, we also derive a new closed formula for the Fujiki constants of the Chern classes of tangent bundles of both Hilbert schemes of points on $K3$ surfaces and generalized Kummer varieties. |
| title | Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds |
| topic | Algebraic Geometry High Energy Physics - Theory |
| url | https://arxiv.org/abs/2201.10878 |