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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.12287 |
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| _version_ | 1866914902436741120 |
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| author | Liu, Yongqiang |
| author_facet | Liu, Yongqiang |
| contents | Let X be a complex smooth quasi-projective variety with an epimorphism $ν\colon π_1(X)\twoheadrightarrow \mathbb{Z}^n$. We survey recent developments about the asymptotic behaviour of Betti numbers with any field coefficients and the order of the torsion part of singular integral homology of finite abelian covers of $X$ associated to $ν$, known as the $L^2$-type invariants. We give relations between $L^2$-type invariants, Alexander invariants and cohomology jump loci. When $ν$ is orbifold effective, we give explicit formulas for $L^2$-invariants at homological degree one in terms of geometric information of $X$. We also propose several related open questions for hyperplane arrangement complement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12287 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^2$-type invariants for complex smooth quasi-projective varieties -- a survey Liu, Yongqiang Algebraic Geometry Let X be a complex smooth quasi-projective variety with an epimorphism $ν\colon π_1(X)\twoheadrightarrow \mathbb{Z}^n$. We survey recent developments about the asymptotic behaviour of Betti numbers with any field coefficients and the order of the torsion part of singular integral homology of finite abelian covers of $X$ associated to $ν$, known as the $L^2$-type invariants. We give relations between $L^2$-type invariants, Alexander invariants and cohomology jump loci. When $ν$ is orbifold effective, we give explicit formulas for $L^2$-invariants at homological degree one in terms of geometric information of $X$. We also propose several related open questions for hyperplane arrangement complement. |
| title | $L^2$-type invariants for complex smooth quasi-projective varieties -- a survey |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2406.12287 |