Collapsing of negative Kahler-Einstein metrics

In this paper, we study the collapsing behaviour of negative Kahler-Einstein metrics along degenerations of canonical polarized manifolds. We prove that for a toroidal degeneration of canonical polarized manifolds with the total space Q-factorial, the Kahler-Einstein metrics on fibers collapse to a lower dimensional complete Riemannian manifold in the pointed Gromov-Hausdorff sense by suitably choosing the base points. Furthermore, the most collapsed limit is a real affine Kahler manifold.