On the geometry of Kähler--Frobenius manifolds and their classification
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909447050231808 |
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| author | Combe, Noémie. C. |
| author_facet | Combe, Noémie. C. |
| contents | The purpose of this article is to show that flat compact Kähler manifolds exhibit the structure of a Frobenius manifold, a structure originating in 2D Topological Quantum Field Theory and closely related to Joyce structure. As a result, we classify all such manifolds. It can be deduced that Kähler--Frobenius manifolds include certain Calabi--Yau manifolds, complex tori $T=\mathbb{C}^n/\mathbb{Z}^n$, generalized (orientable) Hantzsche--Wendt manifolds, hyperelliptic manifolds and manifolds of type $T/G$, where $G$ is a finite group acting on $T$ freely and containing no translations. An explicit study is provided for the two-dimensional case. Additionally, we can prove that Chern's conjecture for Kähler pre-Frobenius manifolds holds. Lastly, we establish that certain classes of Kähler-Frobenius manifolds share a direct relationship with theta functions which are important objects in number theory as well as complex analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_14362 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the geometry of Kähler--Frobenius manifolds and their classification Combe, Noémie. C. Differential Geometry Algebraic Geometry Primary: 53A15, 53B05, 53C07, 53C55, 53D45. Secondary: 14K25 The purpose of this article is to show that flat compact Kähler manifolds exhibit the structure of a Frobenius manifold, a structure originating in 2D Topological Quantum Field Theory and closely related to Joyce structure. As a result, we classify all such manifolds. It can be deduced that Kähler--Frobenius manifolds include certain Calabi--Yau manifolds, complex tori $T=\mathbb{C}^n/\mathbb{Z}^n$, generalized (orientable) Hantzsche--Wendt manifolds, hyperelliptic manifolds and manifolds of type $T/G$, where $G$ is a finite group acting on $T$ freely and containing no translations. An explicit study is provided for the two-dimensional case. Additionally, we can prove that Chern's conjecture for Kähler pre-Frobenius manifolds holds. Lastly, we establish that certain classes of Kähler-Frobenius manifolds share a direct relationship with theta functions which are important objects in number theory as well as complex analysis. |
| title | On the geometry of Kähler--Frobenius manifolds and their classification |
| topic | Differential Geometry Algebraic Geometry Primary: 53A15, 53B05, 53C07, 53C55, 53D45. Secondary: 14K25 |
| url | https://arxiv.org/abs/2411.14362 |