On examples of duals Saito's basis of some inhomogeneous divisors, and application
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918293923692544 |
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| author | Kari, Kamtila Dongho, Joseph Assandje, Prosper Rosaire Mama Bouetou, Thomas Bouetou |
| author_facet | Kari, Kamtila Dongho, Joseph Assandje, Prosper Rosaire Mama Bouetou, Thomas Bouetou |
| contents | We investigate a class of non-quasi-homogeneous free divisors in the sense of Saito. These divisors are defined by equations of the form $D:= \{h=0\}$ on $\mathbb{C}^p$, where the polynomial $h$ is specific linear combination of monomials involving the product of coordinates. For this class, we explicitly construct a Saito basis for the module of logarithmic vector fields $Der(logD)$. This construction is then applied to the setting of logarithmic Poisson geometry. Focusing on the example defined by $h=xy+x^{2}y^{2}+x^3y^3$ on the Poisson algebra $(\mathcal{A}=\mathbb{C}[x,y], \{-,-\}_{h})$, where the Poisson bracket is induced by the bivector $π= h\partial x\wedge\partial y$. We define the associated Koszul bracket on the module of logarithmic 1-forms. This enables us to prove that $π$ endows the sheaf of logarithmic 1-forms $Ω^{1}(log D )$ with a Lie-Rinehart algebra structure. Furthermore, we introduce and provide explicit descriptions for the resulting cohomology theory, which we term the logarithmic Poisson cohomology $H_{log}^{\bullet} $ of $\{-,-\}_{h}$. As a related and foundational computation, we also calculate the corresponding logarithmic De Rham cohomology $H^{\bullet}_{DR}$ for the divisor $D$ and we make a generalization in dimension 2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11992 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On examples of duals Saito's basis of some inhomogeneous divisors, and application Kari, Kamtila Dongho, Joseph Assandje, Prosper Rosaire Mama Bouetou, Thomas Bouetou Differential Geometry We investigate a class of non-quasi-homogeneous free divisors in the sense of Saito. These divisors are defined by equations of the form $D:= \{h=0\}$ on $\mathbb{C}^p$, where the polynomial $h$ is specific linear combination of monomials involving the product of coordinates. For this class, we explicitly construct a Saito basis for the module of logarithmic vector fields $Der(logD)$. This construction is then applied to the setting of logarithmic Poisson geometry. Focusing on the example defined by $h=xy+x^{2}y^{2}+x^3y^3$ on the Poisson algebra $(\mathcal{A}=\mathbb{C}[x,y], \{-,-\}_{h})$, where the Poisson bracket is induced by the bivector $π= h\partial x\wedge\partial y$. We define the associated Koszul bracket on the module of logarithmic 1-forms. This enables us to prove that $π$ endows the sheaf of logarithmic 1-forms $Ω^{1}(log D )$ with a Lie-Rinehart algebra structure. Furthermore, we introduce and provide explicit descriptions for the resulting cohomology theory, which we term the logarithmic Poisson cohomology $H_{log}^{\bullet} $ of $\{-,-\}_{h}$. As a related and foundational computation, we also calculate the corresponding logarithmic De Rham cohomology $H^{\bullet}_{DR}$ for the divisor $D$ and we make a generalization in dimension 2. |
| title | On examples of duals Saito's basis of some inhomogeneous divisors, and application |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2601.11992 |