LIFTS OF SMOOTH GROUP ACTIONS TO LINE BUNDLES

Abstract

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 H²_G(X; ℤ), and that the different lifts of the action are classified by the lifts of _c1(L) to H²_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 which has a G-invariant curvature, then there is a lift of the action to a certain power L^d (where d is independent of L) which leaves fixed the induced metric on L^d and the connection ∇^{[otimes ]d}. This generalises to symplectic geometry a well-known result in geometric invariant theory.