On strong Euler-homogeneity and Saito-holonomicity for complex hypersurfaces. Applications to a conjecture on free divisors
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918333124706304 |
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| author | Rodríguez, Abraham del Valle |
| author_facet | Rodríguez, Abraham del Valle |
| contents | We first develop some criteria for a general divisor to be strongly Euler-homogeneous in terms of the Fitting ideals of certain modules. We also study new variants of Saito-holonomicity, generalizing Koszul-free type properties and characterizing them in terms of the same Fitting ideals.
Thanks to these advances, we are able to make progress in the understanding of a conjecture from 2002: a free divisor satisfying the Logarithmic Comparison Theorem (LCT) must be strongly Euler-homogeneous. Previously, it was known to be true only for ambient dimension $n \leq 3$ or assuming Koszul-freeness. We prove it in the following new cases: assuming strong Euler-homogeneity outside a discrete set of points; assuming the divisor is weakly Koszul-free; for $n=4$; for linear free divisors in $n=5$.
Finally, we refute a conjecture stating that all linear free divisors satisfy LCT, are strongly Euler-homogeneous and have $b$-functions with symmetric roots about $-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21829 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On strong Euler-homogeneity and Saito-holonomicity for complex hypersurfaces. Applications to a conjecture on free divisors Rodríguez, Abraham del Valle Algebraic Geometry Complex Variables 32S25 (Primary), 32S05, 14F40 (Secondary) We first develop some criteria for a general divisor to be strongly Euler-homogeneous in terms of the Fitting ideals of certain modules. We also study new variants of Saito-holonomicity, generalizing Koszul-free type properties and characterizing them in terms of the same Fitting ideals. Thanks to these advances, we are able to make progress in the understanding of a conjecture from 2002: a free divisor satisfying the Logarithmic Comparison Theorem (LCT) must be strongly Euler-homogeneous. Previously, it was known to be true only for ambient dimension $n \leq 3$ or assuming Koszul-freeness. We prove it in the following new cases: assuming strong Euler-homogeneity outside a discrete set of points; assuming the divisor is weakly Koszul-free; for $n=4$; for linear free divisors in $n=5$. Finally, we refute a conjecture stating that all linear free divisors satisfy LCT, are strongly Euler-homogeneous and have $b$-functions with symmetric roots about $-1$. |
| title | On strong Euler-homogeneity and Saito-holonomicity for complex hypersurfaces. Applications to a conjecture on free divisors |
| topic | Algebraic Geometry Complex Variables 32S25 (Primary), 32S05, 14F40 (Secondary) |
| url | https://arxiv.org/abs/2504.21829 |