  We discuss the geometry and topology of the complete, non-compact, Ricci-flat
Stenzel metric, on the tangent bundle of S^{n+1}. We obtain explicit results
for all the metrics, and show how they can be obtained from first-order
equations derivable from a superpotential. We then provide an explicit
construction for the harmonic self-dual (p,q)-forms in the middle dimension
p+q=(n+1) for the Stenzel metrics in 2(n+1) dimensions. Only the (p,p)-forms
are L^2-normalisable, while for (p,q)-forms the degree of divergence grows with
|p-q|. We also construct a set of Ricci-flat metrics whose level surfaces are
U(1) bundles over a product of N Einstein-Kahler manifolds, and we construct
examples of harmonic forms there. As an application, we construct new examples
of deformed supersymmetric non-singular M2-branes with such 8-dimensional
transverse Ricci-flat spaces. We show explicitly that the fractional D3-branes
on the 6-dimensional Stenzel metric found by Klebanov and Strassler is
supported by a pure (2,1)-form, and thus it is supersymmetric, while the
example of Pando Zayas-Tseytlin is supported by a mixture of (1,2) and (2,1)
forms. We comment on the implications for the corresponding dual field theories
of our resolved brane solutions.
