On examples of duals Saito's basis of some inhomogeneous divisors, and application

Abstract

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 $h=xy+x^2{y}^2+x^3{y}^3$ on the Poisson algebra $(\mathcal{A}=\mathbb{C}[x,y],\{-,-{\}}_h)$, where the Poisson bracket is induced by the bivector $\pi =h\partial x\wedge \partial y$. We define the associated Koszul bracket on the module of logarithmic 1-forms. This enables us to prove that $\pi$ endows the sheaf of logarithmic 1-forms ${\Omega }^1(logD)$ 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 $H_{DR}^{\bullet }$ for the divisor $D$ and we make a generalization in dimension 2.