  In this paper, we address the issue of linear stability of Schwarzschild
space- time subject to certain axisymmetric perturbations. In particular, we
prove that associ- ated solutions to the linearized vacuum Einstein equations
centered at a Schwarzschild metric, with suitably regular initial data, decay
to a linearized Kerr metric. Our method employs a complex line bundle
interpretation applied to a connection-level object, allow- ing for direct
analysis of this connection-level object by the linearized Einstein equations,
in contrast with the recent breakthrough of Dafermos-Holzegel-Rodnianski.
