  We study cohomological gauge theories on total spaces of holomorphic line
bundles over complex manifolds and obtain their reduction to the base manifold
by U(1) equivariant localization of the path integral. We exemplify this
general mechanism by proving via exact path integral localization a reduction
for local curves conjectured in hep-th/0411280, relevant to the calculation of
black hole entropy/Gromov-Witten invariants. Agreement with the
four-dimensional gauge theory is recovered by taking into account in the latter
non-trivial contributions coming from one-loop fluctuations determinants at the
boundary of the total space. We also study a class of abelian gauge theories on
Calabi-Yau local surfaces, describing the quantum foam for the A-model,
relevant to the calculation of Donaldson-Thomas invariants.
