On Euler-homogeneity for free divisors
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912353955610624 |
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| author | Rodríguez, Abraham del Valle |
| author_facet | Rodríguez, Abraham del Valle |
| contents | In 2002, it was conjectured that a free divisor satisfying the so-called Logarithmic Comparison Theorem must be strongly Euler-homogeneous and it was proved for the two-dimensional case. Later, in 2006, it was shown that the conjecture is also true in dimension three, but, today, the answer for the general case remains unknown. In this paper, we use the decomposition of a singular derivation as the sum of a semisimple and a topologically nilpotent derivation that commute in order to deal with this problem. By showing that this decomposition preserves the property of being logarithmic, we are able to give alternative proofs for the low-dimensional known cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_11277 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Euler-homogeneity for free divisors Rodríguez, Abraham del Valle Algebraic Geometry Complex Variables 32C38, 14F10, 14F40 In 2002, it was conjectured that a free divisor satisfying the so-called Logarithmic Comparison Theorem must be strongly Euler-homogeneous and it was proved for the two-dimensional case. Later, in 2006, it was shown that the conjecture is also true in dimension three, but, today, the answer for the general case remains unknown. In this paper, we use the decomposition of a singular derivation as the sum of a semisimple and a topologically nilpotent derivation that commute in order to deal with this problem. By showing that this decomposition preserves the property of being logarithmic, we are able to give alternative proofs for the low-dimensional known cases. |
| title | On Euler-homogeneity for free divisors |
| topic | Algebraic Geometry Complex Variables 32C38, 14F10, 14F40 |
| url | https://arxiv.org/abs/2305.11277 |