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 $h=xy+x^{2}y^{2}+x^3y^3$ 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 $\Omega^{1}(log D )$ 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.