Crepant resolutions of ℂ3∕Z4 and the generalized Kronheimer construction (in view of the gauge/gravity correspondence)

As a continuation of a general program started in two previous publications, in the present paper we study the Kähler quotient resolution of the orbifold c3/z4 , 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 F2 . We determine the explicit Kähler geometry of the smooth resolved manifold Y , which is the total space of the canonical bundle of F2. 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 P[1,1,2] , while the other is the product of the ALE space A1 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.