Bounding Codimension-One Subvarieties and a General Inequality between Chern Numbers

We extend the Miyaoka-Yau inequality for a surface to an arbitrary nonuniruled normal complex projective variety, eliminating the hypothesis that the variety must be minimal. The inequality is sharp in dimension three and is also sharp among minimal varieties. For nonminimal varieties in dimension four or higher, an error term is picked up which can be controlled. As a consequence, we bound codimension one subvarieties in a variety of general type linearly in terms of their Chern numbers. In particular, we show that there are only a finite number of smooth Fano, Abelian and Calabi-Yau subvarieties of codimension one in any variety of general type.