Complex geometry of conifolds and a 5-brane wrapped on a 2-sphere

We investigate solutions of type II supergravity which have the product ^1,3×M⁶ structure with non-compact M⁶ factor and which preserve at least four supersymmetries. In particular, we consider various conifolds and the N = 1 supersymmetric `NS5-brane wrapped on a 2-sphere' solution recently discussed in hep-th/0008001. In all of these cases, we explicitly construct the complex structures, and the Kähler and parallel (3,0)-forms of the corresponding M⁶. In addition, we verify that the above solutions preserve, respectively, eight and four supersymmetries of the underlying type II theory. We also demonstrate that the ordinary and fractional D3-brane (5-brane wrapped on a 2-cycle) solutions on singular, resolved and deformed conifolds, and the (S-dual of) NS5-brane wrapped on 2-sphere can be obtained as special cases from a universal ansatz for the supergravity fields, i.e. from a single one-dimensional action governing their radial evolution. We show that like the 3-branes on conifolds, the NS5-brane on a 2-sphere background can be found as a solution of a first-order system following from a superpotential.