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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.01554 |
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| _version_ | 1866917709852180480 |
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| author | Alhwaimel, Mazen M |
| author_facet | Alhwaimel, Mazen M |
| contents | For a line bundle $L$ on a smooth projective surface $X$ and nonnegative integers $k_1, \ldots, k_N$, Okounkov \cite{Oko} introduced the reduced generating series $\big \langle {\rm ch}_{k_1}^{L} \cdots {\rm ch}_{k_N}^{L} \big \rangle'$ for the intersection numbers among the Chern characters of the tautological bundles over the Hilbert schemes of points on $X$ and the total Chern classes of the tangent bundles of these Hilbert schemes. In \cite{Qin2}, Qin conjectured that these reduced generating series are quasi-modular forms if the canonical divisor of $X$ is numerically trivial. In this paper, we verify that Qin's conjecture holds for $\langle {\rm ch}_1^{L_1}{\rm ch}_1^{L_2} \rangle'$. The main approaches are to use the methods laid out in \cite{QY} and construct various relations regarding multiple $q$-zeta values and quasi-modular forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01554 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toward Qin's Conjecture on Hilbert schemes of points and quasi-modular forms Alhwaimel, Mazen M Algebraic Geometry For a line bundle $L$ on a smooth projective surface $X$ and nonnegative integers $k_1, \ldots, k_N$, Okounkov \cite{Oko} introduced the reduced generating series $\big \langle {\rm ch}_{k_1}^{L} \cdots {\rm ch}_{k_N}^{L} \big \rangle'$ for the intersection numbers among the Chern characters of the tautological bundles over the Hilbert schemes of points on $X$ and the total Chern classes of the tangent bundles of these Hilbert schemes. In \cite{Qin2}, Qin conjectured that these reduced generating series are quasi-modular forms if the canonical divisor of $X$ is numerically trivial. In this paper, we verify that Qin's conjecture holds for $\langle {\rm ch}_1^{L_1}{\rm ch}_1^{L_2} \rangle'$. The main approaches are to use the methods laid out in \cite{QY} and construct various relations regarding multiple $q$-zeta values and quasi-modular forms. |
| title | Toward Qin's Conjecture on Hilbert schemes of points and quasi-modular forms |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2407.01554 |