Modular classes of Jacobi bundles

This paper is the first part of a dilogy devoted to modular classes of Jacobi structures from the general line bundle perspective as well as their associated Lie algebroids. First, we explain the relationship between Jacobi algebroids and their associated Gerstenhaber-Jacobi algebras. Then, we show that given a Jacobi manifold, there is a differential complex associated to it whose differential operator is similar to the so-called Koszul-Brylinski operator. This allows us to define Jacobi homology for Jacobi bundles. Moreover, we show that there are generating operators for the Gerstenhaber-Jacobi algebra associated to the Atiyah algebroid DL whose sections are derivations of the associated line bundle L.