Crepant resolutions of ℂ3∕Z4 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 Kahler quotient resolution of the orbifold ℂ 3 ∕ 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 F 2 . We determine the explicit Kahler geometry of the smooth resolved manifold Y , which is the total space of the canonical bundle of 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 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 is a subtler way. These geometrical results will be used to look for exact supergravity brane solutions and dual superconformal gauge theories.