Resolution à la Kronheimer of C3/ Γ singularities and the Monge–Ampère equation for Ricci-flat Kähler metrics in view of D3-brane solutions of supergravity

In this paper, we analyze the relevance of the generalized Kronheimer construction for the gauge/gravity correspondence. We begin with the general structure of D3-brane solutions of type IIB supergravity on smooth manifolds Y-Gamma that are supposed to be the crepant resolution of quotient singularities C-3/Gamma with Gamma a finite subgroup of SU(3). We emphasize that nontrivial 3-form fluxes require the existence of imaginary self-dual harmonic forms omega(2,1). Although excluded in the classical Kronheimer construction, they may be reintroduced by means of mass deformations. Next we concentrate on the other essential item for the D3-brane construction, namely, the existence of a Ricci-flat metric on Y-Gamma. We study the issue of Ricci-flat Kahler metrics on such resolutions Y-Gamma, with particular attention to the case Gamma = Z(4). We advance the conjecture that on the exceptional divisor of Y-Gamma the Kronheimer Kahler 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 tot K-WP[112] that we construct, i.e., the total space of the canonical bundle of the weighted projective space WP[112], which is a partial resolution of C-3/Z(4). For the full resolution, we have Y-Z4 = tot K-F2, where F-2 is the second Hirzebruch surface. We try to extend the proof of the conjecture to this case using the one-parameter Kahler metric on F-2 produced by the Kronheimer construction as initial datum in a Monge-Ampere (MA) equation. We exhibit three formulations of this MA equation, one in terms of the Kahler potential, the other two in terms of the symplectic potential but with two different choices of the variables. In both cases, on