D3-brane supergravity solutions from Ricci-flat metrics on canonical bundles of Kähler–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¯) 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 M6 , 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 Kähler surface MB . This situation emerges in many cases while considering the resolution à la Kronheimer of singular manifolds of type M6= C3/ Γ , where Γ ⊂ SU (3) is a discrete subgroup. When Γ = Z4 , the surface MB is the second Hirzebruch surface endowed with a Kähler metric having SU (2) × U (1) isometry. There is an entire class Met (FV) of such cohomogeneity one Kähler 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 Kähler–Einstein metrics of the aforementioned class, defined on manifolds that are homeomorphic to S2× S2 , but are singular as complex manifolds. Actually, Met (FV) KE⊂ Met (FV) ext⊂ Met (FV) is a subset of a four parameter subclass Met (FV) ext of cohomogeneity one extremal Kähler 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 Kähler–Einstein nature of the metrics in Met (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 in Met (FV) KE are defined on the base manifolds of U(1) fibrations supporting the family of Sasaki–Einstein metrics SEmet 5 introduced by Gauntlett et al. (Adv Theor Math Phys 8:711–734, 2004), and already appeared in Gibbons and Pope (Commun Math Phys 66:267–290, 1979). However, as we show in detail using Weyl tensor polynomial invariants, the six-dimensional Ricci-flat metric on the metric cone of M5∈ Met (SE) 5 is different from the Ricci-flat metric on tot [K[(MKE)] constructed via Calabi Ansatz. This opens new research perspectives. We also show the full integrability of the differential system of geodesics equations on MB thanks to a certain conserved quantity which is similar to the Carter constant in the case of the Kerr metric.