A geometric integration of the extended Lee homomorphism

We give a geometric integration of the extended Lee homomorphism, yielding a homomorphism on the group of automorphisms of a locally conformal symplectic manifold and interpret its kernel as quotient of a group of symplectic diffeomorphisms of a canonically associated symplectic manifold, on which we construct the Calabi invariants in terms of the cA-cohomology. The value of this global Lee homomorphism on an automorphism is the similitude ratio of some lifting on the associated symplectic manifold. Applications to mechanics are given.