Lifts of Smooth Group Actions to Line Bundles

Let X be a compact manifold with a smooth action of a compact connected Lie group G. Let L → X be a complex line bundle. Using the Cartan complex for equivariant cohomology, we give a new proof of a theorem of Hattori and Yoshida which says that the action of G lifts to L if and only if the first Chern class c1(L) of L can be lifted to an integral equivariant cohomology class in H G (X ;Z), and that the different lifts of the action are classified by the lifts of c1(L) to H 2 G (X ;Z). As a corollary of our method of proof, we prove that, if the action is Hamiltonian and ∇ is a connection on L which is unitary for some metric on L and whose curvature is G-invariant, then there is a lift of the action to a certain power L (where d is independent of L) which leaves fixed the induced metric on L and the connection ∇. This generalises to symplectic geometry a well known result in Geometric Invariant Theory.