On the Uniqueness of Kähler-Einstein Polygons in Mutation-Equivalence Classes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913223141228544 |
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| author | Hall, Thomas |
| author_facet | Hall, Thomas |
| contents | We study a subclass of Kähler-Einstein Fano polygons and how they behave under mutation. The polygons of interest are Kähler-Einstein Fano triangles and symmetric Fano polygons. In particular, we find an explicit bound for the number of these polygons in an arbitrary mutation-equivalence class.
An important mutation-invariant of a Fano polygon is its singularity content. We extend the notion of singularity content and prove that it is still a mutation-invariant. We use this to show that if two symmetric Fano polygons are mutation-equivalent, then they are isomorphic. We further show that if two Kähler-Einstein Fano triangles are mutation-equivalent, then they are isomorphic. Finally, we show that if a symmetric Fano polygon is mutation-equivalent to a Kähler-Einstein triangle, then they are isomorphic. Thus, each mutation-equivalence class has at most one Fano polygon which is either a Kähler-Einstein triangle or symmetric.
A recent conjecture states that all Kähler-Einstein Fano polygons are either triangles or are symmetric. We provide a counterexample $P$ to this conjecture and discuss several of its properties. For instance, we compute iterated barycentric transformations of $P$ and find that (a) the Kähler-Einstein property is not preserved by the barycentric transformation, and (b) $P$ is of strict type $B_2$. Finally, we find examples of Kähler-Einstein Fano polygons which are not minimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02832 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Uniqueness of Kähler-Einstein Polygons in Mutation-Equivalence Classes Hall, Thomas Combinatorics Algebraic Geometry 52B99 (Primary), 14J45 (Secondary) We study a subclass of Kähler-Einstein Fano polygons and how they behave under mutation. The polygons of interest are Kähler-Einstein Fano triangles and symmetric Fano polygons. In particular, we find an explicit bound for the number of these polygons in an arbitrary mutation-equivalence class. An important mutation-invariant of a Fano polygon is its singularity content. We extend the notion of singularity content and prove that it is still a mutation-invariant. We use this to show that if two symmetric Fano polygons are mutation-equivalent, then they are isomorphic. We further show that if two Kähler-Einstein Fano triangles are mutation-equivalent, then they are isomorphic. Finally, we show that if a symmetric Fano polygon is mutation-equivalent to a Kähler-Einstein triangle, then they are isomorphic. Thus, each mutation-equivalence class has at most one Fano polygon which is either a Kähler-Einstein triangle or symmetric. A recent conjecture states that all Kähler-Einstein Fano polygons are either triangles or are symmetric. We provide a counterexample $P$ to this conjecture and discuss several of its properties. For instance, we compute iterated barycentric transformations of $P$ and find that (a) the Kähler-Einstein property is not preserved by the barycentric transformation, and (b) $P$ is of strict type $B_2$. Finally, we find examples of Kähler-Einstein Fano polygons which are not minimal. |
| title | On the Uniqueness of Kähler-Einstein Polygons in Mutation-Equivalence Classes |
| topic | Combinatorics Algebraic Geometry 52B99 (Primary), 14J45 (Secondary) |
| url | https://arxiv.org/abs/2402.02832 |