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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.13109 |
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| _version_ | 1866917274023100416 |
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| author | Fonteyne, Lise Veys, Willem |
| author_facet | Fonteyne, Lise Veys, Willem |
| contents | Motivic and topological zeta functions are singularity invariants, mainly associated to a function $f$ and a top differential form $ω$ on a smooth variety. When $ω$ is the standard form $dx_1\wedge \dots \wedge dx_n$ on affine $n$-space, the monodromy conjecture states that poles of these zeta functions should induce monodromy eigenvalues of $f$. We study natural generalized statements of the monodromy conjecture for functions $f$ on complex surface germs; more precisely on singular surfaces for forms $ω$ that generalize the standard form, and on the affine plane for forms $ω$ that are intrinsically associated to $f$.
For all cases, we provide counterexamples to the statement. In addition, when the intrinsically associated $ω$ is given by the generic polar of $f$, we discover a relation between the poles of the zeta functions and the intersection behaviour of the polar curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13109 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a Generalized Monodromy Conjecture for Curves using Differential Forms Fonteyne, Lise Veys, Willem Algebraic Geometry Motivic and topological zeta functions are singularity invariants, mainly associated to a function $f$ and a top differential form $ω$ on a smooth variety. When $ω$ is the standard form $dx_1\wedge \dots \wedge dx_n$ on affine $n$-space, the monodromy conjecture states that poles of these zeta functions should induce monodromy eigenvalues of $f$. We study natural generalized statements of the monodromy conjecture for functions $f$ on complex surface germs; more precisely on singular surfaces for forms $ω$ that generalize the standard form, and on the affine plane for forms $ω$ that are intrinsically associated to $f$. For all cases, we provide counterexamples to the statement. In addition, when the intrinsically associated $ω$ is given by the generic polar of $f$, we discover a relation between the poles of the zeta functions and the intersection behaviour of the polar curve. |
| title | On a Generalized Monodromy Conjecture for Curves using Differential Forms |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2602.13109 |