  We derive a novel deformation of the warped resolved conifold background with
supersymmetry breaking ISD (1,2) fluxes by adding D7-branes to this type IIB
theory. We find spontaneous supersymmetry breaking without generating a bulk
cosmological constant. In the compactified form, our background will no longer
be a Calabi-Yau manifold as it allows a non-vanishing first Chern class. In the
presence of D7-branes the (1,2) fluxes can give rise to non-trivial D-terms. We
study the Ouyang embedding of D7-branes in detail and find that in this case
the D-terms are indeed non-zero. In the limit when we approach the singular
conifold, the D-terms vanish for Ouyang's embedding, although supersymmetry
appears to be broken.
  We also construct the F-theory lift of our background and demonstrate how
these IIB (1,2) fluxes lift to non-primitive (2,2) flux on the fourfold. The
seven branes correspond to normalisable harmonic forms. We briefly sketch a
possible way to attain an inflaton potential in this background once extra
D3-branes are introduced and point out some possibilities of restoring
supersymmetry in our background that could in principle be used as the end
point of the inflationary set-up. In a companion paper we will analyse in
details the inflationary dynamics in this background.
