Revisiting Gamma conjecture I: counterexamples and modifications
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arXiv
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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929676217221120 |
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| author | Galkin, Sergey Hu, Jianxun Iritani, Hiroshi Ke, Huazhong Li, Changzheng Su, Zhitong |
| author_facet | Galkin, Sergey Hu, Jianxun Iritani, Hiroshi Ke, Huazhong Li, Changzheng Su, Zhitong |
| contents | We continue investigation of asymptotics of quantum differential equation for Fano manifolds, with a special regard to Gamma conjecture I and its underlying Conjecture $\mathcal{O}$. We introduce the A-model conifold value, a symplectic invariant of a Fano manifold, and propose modifications for Gamma conjecture I based on this new definition. We discuss an interplay of birational transformations with an extension of Gamma conjecture I over the Kähler moduli space. These heuristics are applied to rigorously identify the principal asymptotic class in the case of $\mathbb{P}^1$-bundles $X_n=\mathbb{P}_{\mathbb{P}^{n}}(\mathcal{O}\oplus\mathcal{O}(n))$. We observe, in particular, that for $X_n$ of dimension at least four, the Conjecture $\mathcal{O}$ holds just for even values of $n$, and in these cases we falsify the original non-modified Gamma conjecture I. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16979 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Revisiting Gamma conjecture I: counterexamples and modifications Galkin, Sergey Hu, Jianxun Iritani, Hiroshi Ke, Huazhong Li, Changzheng Su, Zhitong Algebraic Geometry Mathematical Physics Symplectic Geometry We continue investigation of asymptotics of quantum differential equation for Fano manifolds, with a special regard to Gamma conjecture I and its underlying Conjecture $\mathcal{O}$. We introduce the A-model conifold value, a symplectic invariant of a Fano manifold, and propose modifications for Gamma conjecture I based on this new definition. We discuss an interplay of birational transformations with an extension of Gamma conjecture I over the Kähler moduli space. These heuristics are applied to rigorously identify the principal asymptotic class in the case of $\mathbb{P}^1$-bundles $X_n=\mathbb{P}_{\mathbb{P}^{n}}(\mathcal{O}\oplus\mathcal{O}(n))$. We observe, in particular, that for $X_n$ of dimension at least four, the Conjecture $\mathcal{O}$ holds just for even values of $n$, and in these cases we falsify the original non-modified Gamma conjecture I. |
| title | Revisiting Gamma conjecture I: counterexamples and modifications |
| topic | Algebraic Geometry Mathematical Physics Symplectic Geometry |
| url | https://arxiv.org/abs/2405.16979 |