Fundamental groups of log Calabi-Yau surfaces
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910820533796864 |
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| author | Gachet, Cécile Liu, Zhining Moraga, Joaquín |
| author_facet | Gachet, Cécile Liu, Zhining Moraga, Joaquín |
| contents | In this article, we study the orbifold fundamental group $π_1^{\rm orb}(X,Δ)$ of a Calabi--Yau pair $(X,Δ)$ with log canonical singularities. We conjecture that the orbifold fundamental group $π_1^{\rm orb}(X,Δ)$ of a $n$-dimensional log Calabi--Yau pair admits a normal solvable subgroup of rank at most $2n$ and index at most $c(n)$. We prove this conjecture in the case that $n=2$. More precisely, for a log Calabi--Yau surface pair $(X,Δ)$ we show that $π_1^{\rm orb}(X,Δ)$ is the extension of a nilpotent group of length at most $2$ and rank at most $4$ by a finite group of order at most $7200$. We also show that the bounds on the nilpotency length, rank, and order of the finite group quotient in this result are sharp. Finally, we provide some necessary criteria for a log Calabi--Yau surface $(X,Δ)$ to have an infinite, or a non virtually abelian orbifold fundamental group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_03981 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fundamental groups of log Calabi-Yau surfaces Gachet, Cécile Liu, Zhining Moraga, Joaquín Algebraic Geometry Primary 14E30, 14F35, Secondary 90C57, 14M25, 20F34 In this article, we study the orbifold fundamental group $π_1^{\rm orb}(X,Δ)$ of a Calabi--Yau pair $(X,Δ)$ with log canonical singularities. We conjecture that the orbifold fundamental group $π_1^{\rm orb}(X,Δ)$ of a $n$-dimensional log Calabi--Yau pair admits a normal solvable subgroup of rank at most $2n$ and index at most $c(n)$. We prove this conjecture in the case that $n=2$. More precisely, for a log Calabi--Yau surface pair $(X,Δ)$ we show that $π_1^{\rm orb}(X,Δ)$ is the extension of a nilpotent group of length at most $2$ and rank at most $4$ by a finite group of order at most $7200$. We also show that the bounds on the nilpotency length, rank, and order of the finite group quotient in this result are sharp. Finally, we provide some necessary criteria for a log Calabi--Yau surface $(X,Δ)$ to have an infinite, or a non virtually abelian orbifold fundamental group. |
| title | Fundamental groups of log Calabi-Yau surfaces |
| topic | Algebraic Geometry Primary 14E30, 14F35, Secondary 90C57, 14M25, 20F34 |
| url | https://arxiv.org/abs/2312.03981 |