Fundamental groups, coregularity, and low dimensional klt Calabi-Yau pairs
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| Format: | Preprint |
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2024
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| _version_ | 1866913358079328256 |
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| author | Braun, Lukas Figueroa, Fernando |
| author_facet | Braun, Lukas Figueroa, Fernando |
| contents | In this article, we study how the absolute coregularity of a projective log pair reflects on its fundamental group. More precisely, we conjecture that for a projective klt log pair $(X,D)$ of absolute coregularity $c$ (and arbitrary dimension) the fundamental group $π_1^{\rm reg}(X,D)$ admits a normal abelian subgroup of finite index and rank at most $2c$. We prove this conjecture in the cases $0 \leq c \leq 3$, building on the almost abelianity of the fundamental groups of klt Calabi-Yau pairs of dimension $\leq 3$. In the cases $c \in \{0,1,2\}$ and fixed dimension, we can furthermore bound the index of a solvable normal subgroup. In dimension three, we are able to prove almost abelianity of the fundamental group of the regular locus for projective klt Calabi-Yau pairs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_01315 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fundamental groups, coregularity, and low dimensional klt Calabi-Yau pairs Braun, Lukas Figueroa, Fernando Algebraic Geometry 14F35, 14E30 In this article, we study how the absolute coregularity of a projective log pair reflects on its fundamental group. More precisely, we conjecture that for a projective klt log pair $(X,D)$ of absolute coregularity $c$ (and arbitrary dimension) the fundamental group $π_1^{\rm reg}(X,D)$ admits a normal abelian subgroup of finite index and rank at most $2c$. We prove this conjecture in the cases $0 \leq c \leq 3$, building on the almost abelianity of the fundamental groups of klt Calabi-Yau pairs of dimension $\leq 3$. In the cases $c \in \{0,1,2\}$ and fixed dimension, we can furthermore bound the index of a solvable normal subgroup. In dimension three, we are able to prove almost abelianity of the fundamental group of the regular locus for projective klt Calabi-Yau pairs. |
| title | Fundamental groups, coregularity, and low dimensional klt Calabi-Yau pairs |
| topic | Algebraic Geometry 14F35, 14E30 |
| url | https://arxiv.org/abs/2401.01315 |