Generalized conifolds and four dimensional $N=1$ superconformal theories

We consider D3-branes placed at singularities of Calabi-Yau threefolds which generalize the conifold singularity and have an ADE classification. The $\CN=1$ superconformal theories dictating their low-energy dynamics are infrared fixed points arising from deforming the corresponding ADE $\CN=2$ superconformal field theories by mass terms for adjoint chiral fields. We probe the geometry by a single $D3$-brane and discuss the near-horizon supergravity solution for a large number $N$ of coincident $D3$-branes.