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We analyse the (rigid) special geometry of a class of local Calabi-Yaumanifolds given by hypersurfaces in C^4 as W'(x)^2+f_0(x)+v^2+w^2+z^2=0, thatarise in the study of the large N duals of four-dimensional N=1 supersymmetricSU(N) Yang-Mills theories with adjoint field \Phi and superpotential W(\Phi).The special geometry relations are deduced from the planar limit of thecorresponding holomorphic matrix model. The set of cycles is split into a bulksector, for which we obtain the standard rigid special geometry relations, anda set of relative cycles, that come from the non-compactness of the manifold,for which we find cut-off dependent corrections to the usual special geometryrelations. The (cut-off independent) prepotential is identified with the(analytically continued) free energy of the holomorphic matrix model in theplanar limit. On the way, we clarify various subtleties pertaining to thesaddle point approximation of the holomorphic matrix model. A formula for thesuperpotential of IIB string theory with background fluxes on these localCalabi-Yau manifolds is proposed that is based on pairings similar to the onesof relative cohomology.Comment: 40 pages, 4 figure

Special geometry of local Calabi-Yau manifolds and superpotentials from holomorphic matrix models