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 YΓ that are supposed to be the crepant resolution of quotient singularities C3/ Γ with Γ a finite subgroup of SU(3). We emphasize that nontrivial 3-form fluxes require the existence of imaginary self-dual harmonic forms ω2 , 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 YΓ. We study the issue of Ricci-flat Kähler metrics on such resolutions YΓ, with particular attention to the case Γ = Z4. We advance the conjecture that on the exceptional divisor of YΓ 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 tot KWP[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 C3/ Z4. For the full resolution, we have YZ4=totKF2, where F2 is the second Hirzebruch surface. We try to extend the proof of the conjecture to this case using the one-parameter Kähler metric on F2 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 but with two different choices of the variables. In both cases, one can establish a series solution in powers of the variable along the fibers 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, rather it uniquely determines all the subsequent terms as local functionals of this initial datum. Although 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.