In this paper, we study the collapsing behaviour of negative K\"{ a}hler-Einstein metrics along degenerations of canonical polarized manifolds. We prove that for a toroidal degeneration of canonical polarized manifolds with the total space $\mathbb{ Q}$-factorial, the K\"{ a}hler-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 K\"{ a}hler manifold.
Categories: math.DG math.AG math.MG
