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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2508.01374 |
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| _version_ | 1866911291286749184 |
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| author | Lee, Shuang-Yen Wang, Chin-Lung Wang, Sz-Sheng |
| author_facet | Lee, Shuang-Yen Wang, Chin-Lung Wang, Sz-Sheng |
| contents | A threefold extremal transition $Y \searrow X$ consists of a crepant extremal contraction $ϕ\colon Y \to \bar Y$ with curve class $\ell \in \operatorname{NE}(Y)$, followed by a smoothing $\bar Y\rightsquigarrow X$. We consider the Type II case that $ϕ$ contracts a divisor $E$ to a point and prove that the quantum cohomology $QH(X)$ is obtained from $QH(Y)$ via analytic continuation, regularization, and specialization in $Q^\ell$. Besides roots of unity, special $\mathrm{L}$-values appear in $\lim Q^\ell$ whenever $\bar Y$ admits more than one smoothings.
Further techniques are employed and explored beyond known tools in Gromov--Witten theory including (i) the canonical local B model attached to $Y \searrow X$, (ii) existence of semistable reduction of double point type for the smoothing, (iii) the modularity of the extremal function $\mathbb{E} := E^3/\langle E, E, E\rangle^Y$, and (iv) periods integrals of Eisenstein series. Our study provides a geometric framework linking classifications of del Pezzo surfaces, Ramanujan's theta functions, and Zagier's special ODE list via Type II transitions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01374 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum Extremal Transitions and Special L-values Lee, Shuang-Yen Wang, Chin-Lung Wang, Sz-Sheng Algebraic Geometry A threefold extremal transition $Y \searrow X$ consists of a crepant extremal contraction $ϕ\colon Y \to \bar Y$ with curve class $\ell \in \operatorname{NE}(Y)$, followed by a smoothing $\bar Y\rightsquigarrow X$. We consider the Type II case that $ϕ$ contracts a divisor $E$ to a point and prove that the quantum cohomology $QH(X)$ is obtained from $QH(Y)$ via analytic continuation, regularization, and specialization in $Q^\ell$. Besides roots of unity, special $\mathrm{L}$-values appear in $\lim Q^\ell$ whenever $\bar Y$ admits more than one smoothings. Further techniques are employed and explored beyond known tools in Gromov--Witten theory including (i) the canonical local B model attached to $Y \searrow X$, (ii) existence of semistable reduction of double point type for the smoothing, (iii) the modularity of the extremal function $\mathbb{E} := E^3/\langle E, E, E\rangle^Y$, and (iv) periods integrals of Eisenstein series. Our study provides a geometric framework linking classifications of del Pezzo surfaces, Ramanujan's theta functions, and Zagier's special ODE list via Type II transitions. |
| title | Quantum Extremal Transitions and Special L-values |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2508.01374 |