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We analyze the relevance of the generalized Kronheimer construction for thegauge-gravity correspondence. We study the general structure of IIBsupergravity D3-brane solutions on crepant resolutions $Y$ of singularities$\mathbb{C}^3/\Gamma$ with $\Gamma$ a finite subgroup of $SU(3)$. Next weconcentrate on another essential item for the D3-brane construction, i.e., theexistence of a Ricci-flat metric on $Y$, with particular attention to the case$\Gamma=\mathbb{Z}_4$. We conjecture that on the exceptional divisor theKronheimer K\"ahler metric and the Ricci-flat one, that is locally flat atinfinity, coincide. The conjecture is shown to be true in the case of theRicci-flat metric on ${\rm tot} K_{{\mathbb WP}[112]}$ that we construct, whichis a partial resolution of $\mathbb{C}^3/\mathbb{Z}_4$. For the full resolutionwe have $Y=\operatorname{tot} K_{\mathbb{F}_{2}}$, where $\mathbb{F}_2$ is thesecond Hizebruch surface. We try to extend the proof of the conjecture to thiscase using the one-parameter K\"ahler metric on $\mathbb{F}_2$ produced by theKronheimer construction as initial datum in a Monge-Amp\`{e}re (MA) equation.We exhibit three formulations of this MA equation, one in terms of the K\"ahlerpotential, the other two in terms of the symplectic potential; in all cases onecan establish a series solution in powers of the fiber variable of thecanonical bundle. The main property of the MA equation is that it does notimpose any condition on the initial geometry of the exceptional divisor, butuniquely determines all the subsequent terms as local functionals of theinitial datum. While a formal proof is still missing, numerical and analyticalresults support the conjecture. As a by-product of our investigation we haveidentified some new properties of this type of MA equations that we believe tobe so far unknown.

letters in mathematical physics/letters in mathematical physics

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