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 \(\mathbb {C}^3/\Gamma \) 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 Kähler metrics on such resolutions \(Y^\Gamma \), with particular attention to the case \(\Gamma =\mathbb {Z}_4\). We advance the conjecture that on the exceptional divisor of \(Y^\Gamma \) 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 \(\mathrm{tot} K_{{\mathbb {W}P}[112]}\) that we construct, i.e., the total space of the canonical bundle of the weighted projective space \({\mathbb {W}P}[112]\), which is a partial resolution of \(\mathbb {C}^3/\mathbb {Z}_4\). For the full resolution, we have \(Y^{\mathbb {Z}_4}={\text {tot}} K_{\mathbb {F}_{2}}\), where \(\mathbb {F}_2\) is the second Hirzebruch surface. We try to extend the proof of the conjecture to this case using the one-parameter Kähler metric on \(\mathbb {F}_2\) 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.

在本文中，我们分析了广义 Kronheimer 构造与规范/重力对应关系的相关性。我们从光滑流形\(Y^\Gamma \)上的 IIB 型超重力D3-膜解的一般结构开始，它应该是商奇点的蠕变分辨率\(\mathbb {C}^3/\Gamma \ )与\(\Gamma \)是SU (3) 的一个有限子群。我们强调非平凡的 3 型通量需要存在假想的自对偶调和形式\(\omega ^{2,1}\). 尽管在经典的 Kronheimer 构造中被排除在外，但它们可能会通过质量变形重新引入。接下来，我们专注于 D3-膜结构的另一个基本项目，即在\(Y^\Gamma \)上存在 Ricci-flat 度量。我们研究了在此类分辨率\(Y^\Gamma \)上的 Ricci-flat Kähler 度量问题，特别注意了\(\Gamma =\mathbb {Z}_4\) 的情况。我们提出了一个猜想，即在\(Y^\Gamma \)的例外因数上，Kronheimer Kähler 度量和 Ricci-flat 度量重合，后者在无穷远处局部平坦。在\(\mathrm{tot} K_{{\mathbb {W}P}[112]}\)上的 Ricci-flat 度量的情况下，该猜想被证明是正确的我们构造的，即加权射影空间\({\mathbb {W}P}[112]\)的正则丛的总空间，它是\(\mathbb {C}^3/ \mathbb {Z}_4\)。对于完整分辨率，我们有\(Y^{\mathbb {Z}_4}={\text {tot}} K_{\mathbb {F}_{2}}\)，其中\(\mathbb {F} _2\)是第二个 Hirzebruch 曲面。我们尝试使用\(\mathbb {F}_2\)上的单参数 Kähler 度量将猜想的证明扩展到这种情况由 Kronheimer 构造产生，作为 Monge-Ampère (MA) 方程中的初始数据。我们展示了这个 MA 方程的三种公式，一种是 Kähler 势，另两种是辛势，但有两种不同的变量选择。在这两种情况下，都可以沿规范丛的纤维建立变量幂的级数解。MA 方程的主要性质是它不对例外因数的初始几何结构强加任何条件，而是将所有后续项唯一确定为该初始数据的局部泛函。尽管仍然缺少正式的证明，但数值和分析结果支持了这一猜想。作为我们调查的副产品，