On Some Cohomological Invariants Associated to a Reduced Normal Crossing Divisor in the Affine Plane

Abstract

This work focuses on the study of cohomological invariants associated with a reduced normal crossing divisor in the affine plane. After determining the classical Poisson cohomology groups and the logarithmic Poisson cohomology groups along the associated ideal, we establish that, for a Poisson bivector attached to the log-symplectic structure under consideration, there exists an isomorphism of vector bundles between the module of logarithmic derivations and the module of logarithmic differential 1-forms relative to the ideal. This allows us to construct a cochain morphism related to the de Rham complex and subsequently compute the de Rham cohomology groups in order to carry out a comparison between the resulting cohomology groups.