Crepant Resolutions of $\mathbb{C}^3/\mathbb{Z}_4$ and the Generalized Kronheimer Construction (in view of the Gauge/Gravity Correspondence)

Abstract: As a continuation of a general program started in two previous publications, in the present paper we study the K\"ahler quotient resolution of the orbifold $\mathbb{C}^3/\mathbb{Z}_4$, comparing with the results of a toric description of the same. In this way we determine the algebraic structure of the exceptional divisor, whose compact component is the second Hirzebruch surface $\mathbb F_2$. We determine the explicit K\"ahler geometry of the smooth resolved manifold $Y$, which is the total space of the canonical bundle of $\mathbb F_2$. We study in detail the chamber structure of the space of stability parameters (corresponding in gauge theory to the Fayet-Iliopoulos parameters) that are involved in the construction of the desingularizations either by generalized Kronheimer quotient, or as algebro-geometric quotients. The walls of the chambers correspond to two degenerations; one is a partial desingularization of the quotient, which is the total space of the canonical bundle of the weighted projective space $\mathbb P[1,1,2]$, while the other is the product of the ALE space $A_1$ by a line, and is related to the full resolution in a subtler way. These geometrical results will be used to look for exact supergravity brane solutions and dual superconformal gauge theories.