On Logarithmic Poisson and De Rham Cohomology Groups of a Class of Inhomogeneous Divisors

Abstract

We study logarithmic Poisson and de Rham cohomologies associated with a class of inhomogeneous free divisors in the affine plane. These divisors arise as inhomogeneous deformations of reduced normal crossing divisors and admit explicit bases for their modules of logarithmic vector fields in the sense of Saito. Using these bases, we construct the corresponding logarithmic Poisson structures and describe explicitly the induced Koszul bracket on logarithmic differential 1-forms within the framework of Lie-Rinehart algebras. We then determine the logarithmic Poisson cochain complex and compute its cohomology for the class under consideration. Furthermore, by means of the logarithmic Spencer complex, we identify the corresponding logarithmic de Rham complex and compute its cohomology. These computations provide explicit cohomological invariants for the class of inhomogeneous divisors considered and show how logarithmic Poisson and de Rham theories extend the corresponding constructions for reduced normal crossing divisors. In addition, we establish an explicit comparison between the logarithmic cohomological theories: they are naturally isomorphic in any degree $k \neq 1$, whereas in degree 1 the logarithmic de Rham cohomology group is a split one-dimensional extension of the logarithmic Poisson cohomology group.