Lifts of smooth group actions to line bundles

TL;DR

This paper provides a new proof for when a compact Lie group action lifts to a line bundle over a manifold, using equivariant cohomology, and explores implications for Hamiltonian actions and geometric invariant theory.

Contribution

It introduces a novel proof technique for lifting group actions to line bundles via equivariant cohomology and extends results to Hamiltonian actions in symplectic geometry.

Findings

Lifting of group actions characterized by equivariant cohomology classes.

Existence of a power of the line bundle with a fixed metric and connection under Hamiltonian actions.

Generalization of a known GIT result to symplectic geometry.

Abstract

Let X be a compact manifold with a smooth action of a compact connected Lie group G. Let $L\to X$ 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 $c\sb 1(L)$ of L can be lifted to an integral equivariant cohomology class in $H\sp 2\sb G(X;\ZZ)$, and that the different lifts of the action are classified by the lifts of $c\sb 1(L)$ to $H\sp 2\sb G(X;\ZZ)$. As a corollary of our method of proof, we prove that, if the action is Hamiltonian and $\nabla$ 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 $L\sp d$ (where d is independent of L) which leaves fixed the induced metric on $L^d$ and the connection $\nabla\sp{\otimes d}$. This generalises to symplectic geometry a well known result in Geometric Invariant Theory.