Complex Geometry of Matrix Models

The paper contains some new results and a review of recent achievements concerning multisupport solutions to matrix models. In the leading order of the 't Hooft expansion for matrix integral, these solutions are described by semiclassical, or generalized Whitham, hierarchies and are directly related to the superpotentials of four-dimensional ${\mathcal N}=1$ SUSY gauge theories. We study the derivatives of tau-functions for these solutions associated with families of Riemann surfaces (with possible double points) and find that they satisfy the Witten–Dijkgraaf–Verlinde–Verlinde equations. We also find the free energy in the subleading order in the matrix size and prove that it satisfies certain determinant relations.