D3-brane supergravity solutions from Ricci-flat metrics on canonical bundles of Kahler-Einstein surfaces

D3 brane solutions of type IIB supergravity can be obtained by means of a classical Ansatz involving a harmonic warp factor, H(y, y ($) over bar) multiplying at power -1/2 the first summand, i.e., the Minkowski metric of the D3 brane world-sheet, and at power 1/2 the second summand, i.e., the Ricci-flat metric on a six-dimensional transverse space M-6, whose complex coordinates y are the arguments of the warp factor. Of particular interest is the case where M6 = tot[ K [(MB)] is the total space of the canonical bundle over a complex Kahler surfaceMB. This situation emerges in many cases while considering the resolution a la Kronheimer of singular manifolds of type M-6 = C-3/ Gamma, where Gamma subset of SU(3) is a discrete subgroup. When Gamma = Z(4), the surface M-B Phi is the second Hirzebruch surface endowed with a Kahler metric having SU(2) x U(1) isometry. There is an entire class Met(FV) of such cohomogeneity one Kahler metrics parameterized by a single function FK( v) that are best described in the Abreu-Martelli-Sparks-Yau (AMSY) symplectic formalism. We study in detail a two-parameter subclass Met(FV) (KE). Met(FV) of Kahler-Einstein metrics of the aforementioned class, defined on manifolds that are homeomorphic to S(2)xS(2), but are singular as complex manifolds. Actually, Met(FV) (KE) subset of Met(FV) (ext). Met(FV) is a subset of a four parameter subclass Met(FV) (ext) of cohomogeneity one extremal Kahler metrics originally introduced by Calabi in 1983 and translated by Abreu into the AMSY action-angle formalism. Met(FV) (ext) contains also a two-parameter subclass Met(FV) (extF2) disjoint from Met(FV) (KE) of extremal smooth metrics on the second Hirzebruch surface that we rederive using constraints on period integrals of the Ricci 2-form. The Kahler-Einstein nature of the metrics inMet(FV) (KE) allows the construction of the Ricci-flat metric on their canonical bundle via the Calabi Ansatz, which we recast in the AMSY formalism deriving some new elegant formulae. The metrics