Resolution a la Kronheimer of C^3/ Gamma singularities and the Monge-Ampere equation for Ricci-flat Kaehler metrics in view of D3-brane solutions of supergravity

We analyze the relevance of the generalized Kronheimer construction for the gauge-gravity correspondence. We study the general structure of IIB supergravity D3-brane solutions on crepant resolutions YY of singularities C^3/ Γ where Γ is a finite subgroup of SU(3) . Next we concentrate on another essential item for the D3-brane construction, i.e., the existence of a Ricci-flat metric on YY, with particular attention to the case Gamma= Z4 . We conjecture that on the exceptional divisor the Kronheimer Kähler metric and the Ricci-flat one, that is locally flat at infinity, coincide. The conjecture is shown to be true in the case of the Ricci-flat metric on the total space of the canonical bundle over either WP112 or the second Hizebruch surface with the metric produced by the Kronheimer construction as initial datum in a Monge-Ampère (MA) equation. We exhibit three formulations of this MA equation, one in terms of the Kähler potential, the other two in terms of the symplectic potential; in all cases one can establish a series solution in powers of the fiber variable of the canonical bundle. The main property of the MA equation is that it does not impose any condition on the initial geometry of the exceptional divisor, but uniquely determines all the subsequent terms as local functionals of the initial datum. While a formal proof is still missing, numerical and analytical results support the conjecture. As a by-product of our investigation we have identified some new properties of this type of MA equations that we believe to be so far unknown.