  Ambi-polar metrics, defined so as to allow the signature to change from +4 to
-4 across hypersurfaces, are a mainstay in the construction of BPS microstate
geometries. This paper elucidates the cohomology of these spaces so as to
simplify greatly the construction of infinite families of fluctuating harmonic
magnetic fluxes. It is argued that such fluxes should come from scalar,
harmonic pre-potentials whose source loci are holomorphic divisors. This
insight is obtained by exploring the Kahler structure of ambi-polar
Gibbons-Hawking spaces and it is shown that differentiating the pre-potentials
with respect to Kahler moduli yields solutions to the BPS equations for the
electric potentials sourced by the magnetic fluxes. This suggests that harmonic
analysis on ambi-polar spaces has a novel, and an extremely rich structure,
that is deeply intertwined with the BPS equations. We illustrate our results
using a family of two-centered solutions.
