Collapsing of negative Kähler–Einstein metrics

In this paper, we study the collapsing behaviour of negative Kähler–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 Kähler–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 Kähler manifold.

Full Text (PDF format)

Published 23 May 2016