Lifts of smooth group actions to line bundles
| dc.creator | Riera, Ignasi Mundet i | |
| dc.date | 2000-02-15 | |
| dc.date.accessioned | 2026-07-07T04:33:53Z | |
| dc.date.available | 2026-07-07T04:33:53Z | |
| dc.description | 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. | |
| dc.description | 12 pages | |
| dc.identifier | https://arxiv.org/abs/math/0002123 | |
| dc.identifier | http://arxiv.org/abs/math/0002123 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/58697 | |
| dc.subject | Differential Geometry | |
| dc.subject | Symplectic Geometry | |
| dc.title | Lifts of smooth group actions to line bundles | |
| dc.type | text |