On examples of duals Saito's basis of some inhomogeneous divisors, and application
Kamtila Kari, Joseph Dongho, Prosper Rosaire Mama Assandje, Thomas Bouetou Bouetou
TL;DR
The paper advances the study of free divisors that are not quasi-homogeneous by constructing explicit Saito bases for Der(log D) and Ω^1(log D), and by applying these to logarithmic Poisson geometry. Centered on the inhomogeneous divisor D = {h = 0} with h = $xy + x^{2}y^{2} + x^{3}y^{3}$ in $A = obreak \\mathbb{C}[x,y]$, it defines a log-Poisson structure via the bivector $\\pi = h \,\\partial_x \,\\wedge \\\partial_y$, introduces the Koszul bracket on $\\Omega^{1}(log D)$, and proves a Lie-Rinehart structure through the logarithmic Hamiltonian map. The authors compute the associated logarithmic Poisson cohomology $H_{log}^{\\bullet}$ and logarithmic De Rham cohomology $H_{DR}^{\\bullet}$, obtaining explicit groups such as $H_{log}^{0} \\ obreak \\simeq \\\mathbb{C}$, $H_{log}^{1} \\ obreak \\simeq \\\mathbb{C} imes (\\mathbb{C} \\\oplus xy\\mathbb{C})$, and $H_{log}^{2} \\ obreak \\simeq \\\mathbb{C} \\\oplus xy\\mathbb{C}$ for the principal case, with analogous results and 2D generalizations discussed. It also analyzes the interplay between logarithmic Poisson and De Rham cohomologies in dimension 2, highlighting the role of the log-symplectic Poisson structure. Overall, the work provides explicit bases and cohomological invariants for inhomogeneous free divisors, enriching the interface between free divisor theory and logarithmic Poisson geometry.
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 $π= h\partial x\wedge\partial y$. 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.